Theorem nchoicelem17 6306

Description: Lemma for nchoice 6309. If the special set of a cardinal is finite, then so is the special set of its T-raising, and there is a calculable relationship between their sizes. Theorem 7.2 of [Specker] p. 974. (Contributed by SF, 19-Mar-2015.)

Assertion

Ref	Expression
nchoicelem17	⊢ (( ≤c We NC ∧ M ∈ NC ∧ ( Spac ‘M) ∈ Fin ) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))

Proof of Theorem nchoicelem17

Dummy variables k m n t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		finnc 6244	. . 3 ⊢ (( Spac ‘M) ∈ Fin ↔ Nc ( Spac ‘M) ∈ Nn )
2		risset 2662	. . . 4 ⊢ ( Nc ( Spac ‘M) ∈ Nn ↔ ∃t ∈ Nn t = Nc ( Spac ‘M))
3		nchoicelem13 6302	. . . . . . 7 ⊢ (M ∈ NC → 1c ≤c Nc ( Spac ‘M))
4	3	ad2antlr 707	. . . . . 6 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) → 1c ≤c Nc ( Spac ‘M))
5		1cnc 6140	. . . . . . . 8 ⊢ 1c ∈ NC
6		fvex 5340	. . . . . . . . 9 ⊢ ( Spac ‘M) ∈ V
7	6	ncelncsi 6122	. . . . . . . 8 ⊢ Nc ( Spac ‘M) ∈ NC
8		dflec2 6211	. . . . . . . 8 ⊢ ((1c ∈ NC ∧ Nc ( Spac ‘M) ∈ NC ) → (1c ≤c Nc ( Spac ‘M) ↔ ∃x ∈ NC Nc ( Spac ‘M) = (1c +c x)))
9	5, 7, 8	mp2an 653	. . . . . . 7 ⊢ (1c ≤c Nc ( Spac ‘M) ↔ ∃x ∈ NC Nc ( Spac ‘M) = (1c +c x))
10		eqtr 2370	. . . . . . . . . . . 12 ⊢ ((t = Nc ( Spac ‘M) ∧ Nc ( Spac ‘M) = (1c +c x)) → t = (1c +c x))
11	10	ancoms 439	. . . . . . . . . . 11 ⊢ (( Nc ( Spac ‘M) = (1c +c x) ∧ t = Nc ( Spac ‘M)) → t = (1c +c x))
12		eqtr2 2371	. . . . . . . . . . . . 13 ⊢ ((t = Nc ( Spac ‘M) ∧ t = (1c +c x)) → Nc ( Spac ‘M) = (1c +c x))
13	12	ex 423	. . . . . . . . . . . 12 ⊢ (t = Nc ( Spac ‘M) → (t = (1c +c x) → Nc ( Spac ‘M) = (1c +c x)))
14	13	adantl 452	. . . . . . . . . . 11 ⊢ (( Nc ( Spac ‘M) = (1c +c x) ∧ t = Nc ( Spac ‘M)) → (t = (1c +c x) → Nc ( Spac ‘M) = (1c +c x)))
15	11, 14	jcai 522	. . . . . . . . . 10 ⊢ (( Nc ( Spac ‘M) = (1c +c x) ∧ t = Nc ( Spac ‘M)) → (t = (1c +c x) ∧ Nc ( Spac ‘M) = (1c +c x)))
16		addceq2 4385	. . . . . . . . . . . . . . . . . 18 ⊢ (k = 1c → (x +c k) = (x +c 1c))
17		addccom 4407	. . . . . . . . . . . . . . . . . 18 ⊢ (x +c 1c) = (1c +c x)
18	16, 17	syl6eq 2401	. . . . . . . . . . . . . . . . 17 ⊢ (k = 1c → (x +c k) = (1c +c x))
19	18	eqeq2d 2364	. . . . . . . . . . . . . . . 16 ⊢ (k = 1c → (t = (x +c k) ↔ t = (1c +c x)))
20	19	rspcev 2956	. . . . . . . . . . . . . . 15 ⊢ ((1c ∈ NC ∧ t = (1c +c x)) → ∃k ∈ NC t = (x +c k))
21	5, 20	mpan 651	. . . . . . . . . . . . . 14 ⊢ (t = (1c +c x) → ∃k ∈ NC t = (x +c k))
22		nnnc 6147	. . . . . . . . . . . . . . . 16 ⊢ (t ∈ Nn → t ∈ NC )
23		dflec2 6211	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ NC ∧ t ∈ NC ) → (x ≤c t ↔ ∃k ∈ NC t = (x +c k)))
24	22, 23	sylan2 460	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ NC ∧ t ∈ Nn ) → (x ≤c t ↔ ∃k ∈ NC t = (x +c k)))
25	24	ancoms 439	. . . . . . . . . . . . . 14 ⊢ ((t ∈ Nn ∧ x ∈ NC ) → (x ≤c t ↔ ∃k ∈ NC t = (x +c k)))
26	21, 25	syl5ibr 212	. . . . . . . . . . . . 13 ⊢ ((t ∈ Nn ∧ x ∈ NC ) → (t = (1c +c x) → x ≤c t))
27	26	adantll 694	. . . . . . . . . . . 12 ⊢ (((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) ∧ x ∈ NC ) → (t = (1c +c x) → x ≤c t))
28		nclenn 6250	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ NC ∧ t ∈ Nn ∧ x ≤c t) → x ∈ Nn )
29	28	3expia 1153	. . . . . . . . . . . . . 14 ⊢ ((x ∈ NC ∧ t ∈ Nn ) → (x ≤c t → x ∈ Nn ))
30	29	ancoms 439	. . . . . . . . . . . . 13 ⊢ ((t ∈ Nn ∧ x ∈ NC ) → (x ≤c t → x ∈ Nn ))
31	30	adantll 694	. . . . . . . . . . . 12 ⊢ (((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) ∧ x ∈ NC ) → (x ≤c t → x ∈ Nn ))
32		nchoicelem16 6305	. . . . . . . . . . . . . . . . . 18 ⊢ {t ∣ ( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))))} ∈ V
33		addceq2 4385	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (t = 0c → (1c +c t) = (1c +c 0c))
34		addcid1 4406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1c +c 0c) = 1c
35	33, 34	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t = 0c → (1c +c t) = 1c)
36	35	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (t = 0c → ( Nc ( Spac ‘m) = (1c +c t) ↔ Nc ( Spac ‘m) = 1c))
37	36	imbi1d 308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t = 0c → (( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ( Nc ( Spac ‘m) = 1c → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
38	37	ralbidv 2635	. . . . . . . . . . . . . . . . . . 19 ⊢ (t = 0c → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ∀m ∈ NC ( Nc ( Spac ‘m) = 1c → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
39	38	imbi2d 307	. . . . . . . . . . . . . . . . . 18 ⊢ (t = 0c → (( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ↔ ( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = 1c → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))))))
40		addceq2 4385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t = n → (1c +c t) = (1c +c n))
41	40	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (t = n → ( Nc ( Spac ‘m) = (1c +c t) ↔ Nc ( Spac ‘m) = (1c +c n)))
42	41	imbi1d 308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t = n → (( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
43	42	ralbidv 2635	. . . . . . . . . . . . . . . . . . 19 ⊢ (t = n → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
44	43	imbi2d 307	. . . . . . . . . . . . . . . . . 18 ⊢ (t = n → (( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ↔ ( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))))))
45		addceq2 4385	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (t = (n +c 1c) → (1c +c t) = (1c +c (n +c 1c)))
46	45	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t = (n +c 1c) → ( Nc ( Spac ‘m) = (1c +c t) ↔ Nc ( Spac ‘m) = (1c +c (n +c 1c))))
47	46	imbi1d 308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (t = (n +c 1c) → (( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ( Nc ( Spac ‘m) = (1c +c (n +c 1c)) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
48	47	ralbidv 2635	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t = (n +c 1c) → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c (n +c 1c)) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
49		fveq2 5329	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (m = k → ( Spac ‘m) = ( Spac ‘k))
50	49	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (m = k → Nc ( Spac ‘m) = Nc ( Spac ‘k))
51	50	eqeq1d 2361	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (m = k → ( Nc ( Spac ‘m) = (1c +c (n +c 1c)) ↔ Nc ( Spac ‘k) = (1c +c (n +c 1c))))
52		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (m = k → Tc m = Tc k)
53	52	fveq2d 5333	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (m = k → ( Spac ‘ Tc m) = ( Spac ‘ Tc k))
54	53	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (m = k → (( Spac ‘ Tc m) ∈ Fin ↔ ( Spac ‘ Tc k) ∈ Fin ))
55	53	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (m = k → Nc ( Spac ‘ Tc m) = Nc ( Spac ‘ Tc k))
56		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ( Nc ( Spac ‘m) = Nc ( Spac ‘k) → Tc Nc ( Spac ‘m) = Tc Nc ( Spac ‘k))
57	50, 56	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (m = k → Tc Nc ( Spac ‘m) = Tc Nc ( Spac ‘k))
58	57	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (m = k → ( Tc Nc ( Spac ‘m) +c 1c) = ( Tc Nc ( Spac ‘k) +c 1c))
59	55, 58	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (m = k → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ↔ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c)))
60	57	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (m = k → ( Tc Nc ( Spac ‘m) +c 2c) = ( Tc Nc ( Spac ‘k) +c 2c))
61	55, 60	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (m = k → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c) ↔ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))
62	59, 61	orbi12d 690	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (m = k → (( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)) ↔ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))
63	54, 62	anbi12d 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (m = k → ((( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))) ↔ (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
64	51, 63	imbi12d 311	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (m = k → (( Nc ( Spac ‘m) = (1c +c (n +c 1c)) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))))
65	64	cbvralv 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c (n +c 1c)) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ∀k ∈ NC ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
66	48, 65	syl6bb 252	. . . . . . . . . . . . . . . . . . 19 ⊢ (t = (n +c 1c) → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ∀k ∈ NC ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))))
67	66	imbi2d 307	. . . . . . . . . . . . . . . . . 18 ⊢ (t = (n +c 1c) → (( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ↔ ( ≤c We NC → ∀k ∈ NC ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))))
68		addceq2 4385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t = x → (1c +c t) = (1c +c x))
69	68	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (t = x → ( Nc ( Spac ‘m) = (1c +c t) ↔ Nc ( Spac ‘m) = (1c +c x)))
70	69	imbi1d 308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t = x → (( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ( Nc ( Spac ‘m) = (1c +c x) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
71	70	ralbidv 2635	. . . . . . . . . . . . . . . . . . 19 ⊢ (t = x → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c x) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
72	71	imbi2d 307	. . . . . . . . . . . . . . . . . 18 ⊢ (t = x → (( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c t) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ↔ ( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c x) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))))))
73		nchoicelem14 6303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((m ∈ NC ∧ Nc ( Spac ‘m) = 1c) → ¬ (m ↑c 0c) ∈ NC )
74	73	ex 423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (m ∈ NC → ( Nc ( Spac ‘m) = 1c → ¬ (m ↑c 0c) ∈ NC ))
75	74	adantl 452	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( ≤c We NC ∧ m ∈ NC ) → ( Nc ( Spac ‘m) = 1c → ¬ (m ↑c 0c) ∈ NC ))
76		nchoicelem9 6298	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc m) = 2c ∨ Nc ( Spac ‘ Tc m) = 3c))
77		id 19	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ( Nc ( Spac ‘ Tc m) = 2c → Nc ( Spac ‘ Tc m) = 2c)
78		2nnc 6168	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2c ∈ Nn
79	77, 78	syl6eqel 2441	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( Nc ( Spac ‘ Tc m) = 2c → Nc ( Spac ‘ Tc m) ∈ Nn )
80		id 19	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ( Nc ( Spac ‘ Tc m) = 3c → Nc ( Spac ‘ Tc m) = 3c)
81		2p1e3c 6157	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2c +c 1c) = 3c
82		peano2 4404	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (2c ∈ Nn → (2c +c 1c) ∈ Nn )
83	78, 82	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2c +c 1c) ∈ Nn
84	81, 83	eqeltrri 2424	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 3c ∈ Nn
85	80, 84	syl6eqel 2441	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( Nc ( Spac ‘ Tc m) = 3c → Nc ( Spac ‘ Tc m) ∈ Nn )
86	79, 85	jaoi 368	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (( Nc ( Spac ‘ Tc m) = 2c ∨ Nc ( Spac ‘ Tc m) = 3c) → Nc ( Spac ‘ Tc m) ∈ Nn )
87		finnc 6244	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (( Spac ‘ Tc m) ∈ Fin ↔ Nc ( Spac ‘ Tc m) ∈ Nn )
88	86, 87	sylibr 203	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( Nc ( Spac ‘ Tc m) = 2c ∨ Nc ( Spac ‘ Tc m) = 3c) → ( Spac ‘ Tc m) ∈ Fin )
89	76, 88	syl 15	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Spac ‘ Tc m) ∈ Fin )
90		1p1e2c 6156	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1c +c 1c) = 2c
91	90	eqeq2i 2363	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( Nc ( Spac ‘ Tc m) = (1c +c 1c) ↔ Nc ( Spac ‘ Tc m) = 2c)
92		addccom 4407	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1c +c 2c) = (2c +c 1c)
93	92, 81	eqtri 2373	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1c +c 2c) = 3c
94	93	eqeq2i 2363	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( Nc ( Spac ‘ Tc m) = (1c +c 2c) ↔ Nc ( Spac ‘ Tc m) = 3c)
95	91, 94	orbi12i 507	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (( Nc ( Spac ‘ Tc m) = (1c +c 1c) ∨ Nc ( Spac ‘ Tc m) = (1c +c 2c)) ↔ ( Nc ( Spac ‘ Tc m) = 2c ∨ Nc ( Spac ‘ Tc m) = 3c))
96	76, 95	sylibr 203	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc m) = (1c +c 1c) ∨ Nc ( Spac ‘ Tc m) = (1c +c 2c)))
97		nchoicelem3 6292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Spac ‘m) = {m})
98	97	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → Nc ( Spac ‘m) = Nc {m})
99		df1c3g 6142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (m ∈ NC → 1c = Nc {m})
100	99	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → 1c = Nc {m})
101	98, 100	eqtr4d 2388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → Nc ( Spac ‘m) = 1c)
102		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ( Nc ( Spac ‘m) = 1c → Tc Nc ( Spac ‘m) = Tc 1c)
103	101, 102	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → Tc Nc ( Spac ‘m) = Tc 1c)
104		tc1c 6166	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Tc 1c = 1c
105	103, 104	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → Tc Nc ( Spac ‘m) = 1c)
106	105	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Tc Nc ( Spac ‘m) +c 1c) = (1c +c 1c))
107	106	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ↔ Nc ( Spac ‘ Tc m) = (1c +c 1c)))
108	105	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Tc Nc ( Spac ‘m) +c 2c) = (1c +c 2c))
109	108	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c) ↔ Nc ( Spac ‘ Tc m) = (1c +c 2c)))
110	107, 109	orbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → (( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)) ↔ ( Nc ( Spac ‘ Tc m) = (1c +c 1c) ∨ Nc ( Spac ‘ Tc m) = (1c +c 2c))))
111	110	3adant1 973	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → (( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)) ↔ ( Nc ( Spac ‘ Tc m) = (1c +c 1c) ∨ Nc ( Spac ‘ Tc m) = (1c +c 2c))))
112	96, 111	mpbird 223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))
113	89, 112	jca 518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ¬ (m ↑c 0c) ∈ NC ) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))
114	113	3expia 1153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( ≤c We NC ∧ m ∈ NC ) → (¬ (m ↑c 0c) ∈ NC → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))))
115	75, 114	syld 40	. . . . . . . . . . . . . . . . . . 19 ⊢ (( ≤c We NC ∧ m ∈ NC ) → ( Nc ( Spac ‘m) = 1c → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))))
116	115	ralrimiva 2698	. . . . . . . . . . . . . . . . . 18 ⊢ ( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = 1c → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))))
117		addccom 4407	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1c +c (n +c 1c)) = ((n +c 1c) +c 1c)
118	117	eqeq2i 2363	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) ↔ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))
119		nchoicelem13 6302	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (k ∈ NC → 1c ≤c Nc ( Spac ‘k))
120	119	ad2antrl 708	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → 1c ≤c Nc ( Spac ‘k))
121		simp2 956	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) → n ∈ Nn )
122		simpr 447	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c)) → Nc ( Spac ‘k) = ((n +c 1c) +c 1c))
123		simpr 447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((n ∈ Nn ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c)) → Nc ( Spac ‘k) = ((n +c 1c) +c 1c))
124		0cnsuc 4402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (n +c 1c) ≠ 0c
125		peano2 4404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (n ∈ Nn → (n +c 1c) ∈ Nn )
126		peano1 4403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ 0c ∈ Nn
127		1cnnc 4409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ 1c ∈ Nn
128		addccan1 4561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((n +c 1c) ∈ Nn ∧ 0c ∈ Nn ∧ 1c ∈ Nn ) → (((n +c 1c) +c 1c) = (0c +c 1c) ↔ (n +c 1c) = 0c))
129	126, 127, 128	mp3an23 1269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((n +c 1c) ∈ Nn → (((n +c 1c) +c 1c) = (0c +c 1c) ↔ (n +c 1c) = 0c))
130	125, 129	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (n ∈ Nn → (((n +c 1c) +c 1c) = (0c +c 1c) ↔ (n +c 1c) = 0c))
131	130	necon3bid 2552	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (n ∈ Nn → (((n +c 1c) +c 1c) ≠ (0c +c 1c) ↔ (n +c 1c) ≠ 0c))
132	124, 131	mpbiri 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (n ∈ Nn → ((n +c 1c) +c 1c) ≠ (0c +c 1c))
133		addcid2 4408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0c +c 1c) = 1c
134	133	neeq2i 2528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((n +c 1c) +c 1c) ≠ (0c +c 1c) ↔ ((n +c 1c) +c 1c) ≠ 1c)
135	132, 134	sylib 188	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (n ∈ Nn → ((n +c 1c) +c 1c) ≠ 1c)
136	135	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((n ∈ Nn ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c)) → ((n +c 1c) +c 1c) ≠ 1c)
137	123, 136	eqnetrd 2535	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((n ∈ Nn ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c)) → Nc ( Spac ‘k) ≠ 1c)
138	121, 122, 137	syl2an 463	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → Nc ( Spac ‘k) ≠ 1c)
139	138	necomd 2600	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → 1c ≠ Nc ( Spac ‘k))
140		brltc 6115	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1c <c Nc ( Spac ‘k) ↔ (1c ≤c Nc ( Spac ‘k) ∧ 1c ≠ Nc ( Spac ‘k)))
141	120, 139, 140	sylanbrc 645	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → 1c <c Nc ( Spac ‘k))
142		nchoicelem15 6304	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((k ∈ NC ∧ 1c <c Nc ( Spac ‘k)) → (k ↑c 0c) ∈ NC )
143	142	ex 423	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (k ∈ NC → (1c <c Nc ( Spac ‘k) → (k ↑c 0c) ∈ NC ))
144	143	ad2antrl 708	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → (1c <c Nc ( Spac ‘k) → (k ↑c 0c) ∈ NC ))
145		df-3an 936	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) ↔ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c)) ∧ (k ↑c 0c) ∈ NC ))
146		ceclnn1 6190	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((2c ∈ Nn ∧ k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (2c ↑c k) ∈ NC )
147	78, 146	mp3an1 1264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (2c ↑c k) ∈ NC )
148	147	3adant2 974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → (2c ↑c k) ∈ NC )
149	148	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (2c ↑c k) ∈ NC )
150		fveq2 5329	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (m = (2c ↑c k) → ( Spac ‘m) = ( Spac ‘(2c ↑c k)))
151	150	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (m = (2c ↑c k) → Nc ( Spac ‘m) = Nc ( Spac ‘(2c ↑c k)))
152	151	eqeq1d 2361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (m = (2c ↑c k) → ( Nc ( Spac ‘m) = (1c +c n) ↔ Nc ( Spac ‘(2c ↑c k)) = (1c +c n)))
153		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (m = (2c ↑c k) → Tc m = Tc (2c ↑c k))
154	153	fveq2d 5333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (m = (2c ↑c k) → ( Spac ‘ Tc m) = ( Spac ‘ Tc (2c ↑c k)))
155	154	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (m = (2c ↑c k) → (( Spac ‘ Tc m) ∈ Fin ↔ ( Spac ‘ Tc (2c ↑c k)) ∈ Fin ))
156	154	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (m = (2c ↑c k) → Nc ( Spac ‘ Tc m) = Nc ( Spac ‘ Tc (2c ↑c k)))
157		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ( Nc ( Spac ‘m) = Nc ( Spac ‘(2c ↑c k)) → Tc Nc ( Spac ‘m) = Tc Nc ( Spac ‘(2c ↑c k)))
158	151, 157	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (m = (2c ↑c k) → Tc Nc ( Spac ‘m) = Tc Nc ( Spac ‘(2c ↑c k)))
159	158	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (m = (2c ↑c k) → ( Tc Nc ( Spac ‘m) +c 1c) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c))
160	156, 159	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (m = (2c ↑c k) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ↔ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c)))
161	158	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (m = (2c ↑c k) → ( Tc Nc ( Spac ‘m) +c 2c) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))
162	156, 161	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (m = (2c ↑c k) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c) ↔ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))
163	160, 162	orbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (m = (2c ↑c k) → (( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)) ↔ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))))
164	155, 163	anbi12d 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (m = (2c ↑c k) → ((( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))) ↔ (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))))
165	152, 164	imbi12d 311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (m = (2c ↑c k) → (( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))))))
166	165	rspccva 2955	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ∧ (2c ↑c k) ∈ NC ) → ( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))))
167		tce2 6237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → Tc (2c ↑c k) = (2c ↑c Tc k))
168	167	fveq2d 5333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → ( Spac ‘ Tc (2c ↑c k)) = ( Spac ‘(2c ↑c Tc k)))
169	168	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ↔ ( Spac ‘(2c ↑c Tc k)) ∈ Fin ))
170	168	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘ Tc (2c ↑c k)) = Nc ( Spac ‘(2c ↑c Tc k)))
171	170	eqeq1d 2361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ↔ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c)))
172	170	eqeq1d 2361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c) ↔ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))
173	171, 172	orbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)) ↔ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))))
174	169, 173	anbi12d 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → ((( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))) ↔ (( Spac ‘(2c ↑c Tc k)) ∈ Fin ∧ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))))
175	174	imbi2d 307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))) ↔ ( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘(2c ↑c Tc k)) ∈ Fin ∧ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))))))
176	175	3adant2 974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → (( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))) ↔ ( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘(2c ↑c Tc k)) ∈ Fin ∧ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))))))
177	176	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))) ↔ ( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘(2c ↑c Tc k)) ∈ Fin ∧ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))))))
178		nchoicelem7 6296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘k) = ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
179	178	3adant2 974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘k) = ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
180		simp2 956	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘k) = ((n +c 1c) +c 1c))
181	179, 180	eqtr3d 2387	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ((n +c 1c) +c 1c))
182	181	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ((n +c 1c) +c 1c))
183		nnnc 6147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((n +c 1c) ∈ Nn → (n +c 1c) ∈ NC )
184		fvex 5340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ( Spac ‘(2c ↑c k)) ∈ V
185	184	ncelncsi 6122	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ Nc ( Spac ‘(2c ↑c k)) ∈ NC
186		peano4nc 6151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (( Nc ( Spac ‘(2c ↑c k)) ∈ NC ∧ (n +c 1c) ∈ NC ) → (( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ((n +c 1c) +c 1c) ↔ Nc ( Spac ‘(2c ↑c k)) = (n +c 1c)))
187	185, 186	mpan 651	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((n +c 1c) ∈ NC → (( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ((n +c 1c) +c 1c) ↔ Nc ( Spac ‘(2c ↑c k)) = (n +c 1c)))
188	125, 183, 187	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (n ∈ Nn → (( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ((n +c 1c) +c 1c) ↔ Nc ( Spac ‘(2c ↑c k)) = (n +c 1c)))
189	188	ad2antlr 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ((n +c 1c) +c 1c) ↔ Nc ( Spac ‘(2c ↑c k)) = (n +c 1c)))
190	182, 189	mpbid 201	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → Nc ( Spac ‘(2c ↑c k)) = (n +c 1c))
191		addccom 4407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (n +c 1c) = (1c +c n)
192	190, 191	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → Nc ( Spac ‘(2c ↑c k)) = (1c +c n))
193		peano2 4404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ( Nc ( Spac ‘(2c ↑c Tc k)) ∈ Nn → ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) ∈ Nn )
194		tccl 6161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (k ∈ NC → Tc k ∈ NC )
195		te0c 6238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (k ∈ NC → ( Tc k ↑c 0c) ∈ NC )
196		nchoicelem7 6296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (( Tc k ∈ NC ∧ ( Tc k ↑c 0c) ∈ NC ) → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
197	194, 195, 196	syl2anc 642	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (k ∈ NC → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
198	197	3ad2ant1 976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
199	198	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
200	199	eleq1d 2419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Nc ( Spac ‘ Tc k) ∈ Nn ↔ ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) ∈ Nn ))
201	193, 200	syl5ibr 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Nc ( Spac ‘(2c ↑c Tc k)) ∈ Nn → Nc ( Spac ‘ Tc k) ∈ Nn ))
202		finnc 6244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (( Spac ‘(2c ↑c Tc k)) ∈ Fin ↔ Nc ( Spac ‘(2c ↑c Tc k)) ∈ Nn )
203		finnc 6244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (( Spac ‘ Tc k) ∈ Fin ↔ Nc ( Spac ‘ Tc k) ∈ Nn )
204	201, 202, 203	3imtr4g 261	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (( Spac ‘(2c ↑c Tc k)) ∈ Fin → ( Spac ‘ Tc k) ∈ Fin ))
205		addceq1 4384	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) → ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) +c 1c))
206		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ( Nc ( Spac ‘k) = ( Nc ( Spac ‘(2c ↑c k)) +c 1c) → Tc Nc ( Spac ‘k) = Tc ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
207	178, 206	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → Tc Nc ( Spac ‘k) = Tc ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
208	207	3adant2 974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → Tc Nc ( Spac ‘k) = Tc ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
209	208	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → Tc Nc ( Spac ‘k) = Tc ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
210		tcdi 6165	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (( Nc ( Spac ‘(2c ↑c k)) ∈ NC ∧ 1c ∈ NC ) → Tc ( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c Tc 1c))
211	185, 5, 210	mp2an 653	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ Tc ( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c Tc 1c)
212	104	addceq2i 4388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ( Tc Nc ( Spac ‘(2c ↑c k)) +c Tc 1c) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c)
213	211, 212	eqtri 2373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ Tc ( Nc ( Spac ‘(2c ↑c k)) +c 1c) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c)
214	209, 213	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → Tc Nc ( Spac ‘k) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c))
215	214	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Tc Nc ( Spac ‘k) +c 1c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) +c 1c))
216	199, 215	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ↔ ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) +c 1c)))
217	205, 216	syl5ibr 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) → Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c)))
218		addceq1 4384	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c) → ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c) +c 1c))
219	214	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Tc Nc ( Spac ‘k) +c 2c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) +c 2c))
220		addc32 4417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) +c 2c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c) +c 1c)
221	219, 220	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Tc Nc ( Spac ‘k) +c 2c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c) +c 1c))
222	199, 221	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c) ↔ ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) = (( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c) +c 1c)))
223	218, 222	syl5ibr 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c) → Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))
224	217, 223	orim12d 811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)) → ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))
225	204, 224	anim12d 546	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ((( Spac ‘(2c ↑c Tc k)) ∈ Fin ∧ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c))) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
226	192, 225	embantd 50	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘(2c ↑c Tc k)) ∈ Fin ∧ ( Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘(2c ↑c Tc k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
227	177, 226	sylbid 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (( Nc ( Spac ‘(2c ↑c k)) = (1c +c n) → (( Spac ‘ Tc (2c ↑c k)) ∈ Fin ∧ ( Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 1c) ∨ Nc ( Spac ‘ Tc (2c ↑c k)) = ( Tc Nc ( Spac ‘(2c ↑c k)) +c 2c)))) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
228	166, 227	syl5 28	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((( ≤c We NC ∧ n ∈ Nn ) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ((∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ∧ (2c ↑c k) ∈ NC ) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
229	228	exp4b 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (( ≤c We NC ∧ n ∈ Nn ) → ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) → ((2c ↑c k) ∈ NC → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))))
230	229	com23 72	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (( ≤c We NC ∧ n ∈ Nn ) → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) → ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → ((2c ↑c k) ∈ NC → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))))
231	230	3impia 1148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) → ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC ) → ((2c ↑c k) ∈ NC → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))))
232	231	imp 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → ((2c ↑c k) ∈ NC → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
233	149, 232	mpd 14	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c) ∧ (k ↑c 0c) ∈ NC )) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))
234	145, 233	sylan2br 462	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ ((k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c)) ∧ (k ↑c 0c) ∈ NC )) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))
235	234	expr 598	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → ((k ↑c 0c) ∈ NC → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
236	144, 235	syld 40	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → (1c <c Nc ( Spac ‘k) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
237	141, 236	mpd 14	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ (k ∈ NC ∧ Nc ( Spac ‘k) = ((n +c 1c) +c 1c))) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c))))
238	237	expr 598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ k ∈ NC ) → ( Nc ( Spac ‘k) = ((n +c 1c) +c 1c) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
239	118, 238	syl5bi 208	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) ∧ k ∈ NC ) → ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
240	239	ralrimiva 2698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( ≤c We NC ∧ n ∈ Nn ∧ ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) → ∀k ∈ NC ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))
241	240	3exp 1150	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ≤c We NC → (n ∈ Nn → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) → ∀k ∈ NC ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))))
242	241	com12 27	. . . . . . . . . . . . . . . . . . 19 ⊢ (n ∈ Nn → ( ≤c We NC → (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) → ∀k ∈ NC ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))))
243	242	a2d 23	. . . . . . . . . . . . . . . . . 18 ⊢ (n ∈ Nn → (( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c n) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))) → ( ≤c We NC → ∀k ∈ NC ( Nc ( Spac ‘k) = (1c +c (n +c 1c)) → (( Spac ‘ Tc k) ∈ Fin ∧ ( Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 1c) ∨ Nc ( Spac ‘ Tc k) = ( Tc Nc ( Spac ‘k) +c 2c)))))))
244	32, 39, 44, 67, 72, 116, 243	finds 4412	. . . . . . . . . . . . . . . . 17 ⊢ (x ∈ Nn → ( ≤c We NC → ∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c x) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))))
245		fveq2 5329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (m = M → ( Spac ‘m) = ( Spac ‘M))
246	245	nceqd 6111	. . . . . . . . . . . . . . . . . . . 20 ⊢ (m = M → Nc ( Spac ‘m) = Nc ( Spac ‘M))
247	246	eqeq1d 2361	. . . . . . . . . . . . . . . . . . 19 ⊢ (m = M → ( Nc ( Spac ‘m) = (1c +c x) ↔ Nc ( Spac ‘M) = (1c +c x)))
248		tceq 6159	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (m = M → Tc m = Tc M)
249	248	fveq2d 5333	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (m = M → ( Spac ‘ Tc m) = ( Spac ‘ Tc M))
250	249	eleq1d 2419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (m = M → (( Spac ‘ Tc m) ∈ Fin ↔ ( Spac ‘ Tc M) ∈ Fin ))
251	249	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (m = M → Nc ( Spac ‘ Tc m) = Nc ( Spac ‘ Tc M))
252		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ( Nc ( Spac ‘m) = Nc ( Spac ‘M) → Tc Nc ( Spac ‘m) = Tc Nc ( Spac ‘M))
253	246, 252	syl 15	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (m = M → Tc Nc ( Spac ‘m) = Tc Nc ( Spac ‘M))
254	253	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (m = M → ( Tc Nc ( Spac ‘m) +c 1c) = ( Tc Nc ( Spac ‘M) +c 1c))
255	251, 254	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (m = M → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ↔ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c)))
256	253	addceq1d 4390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (m = M → ( Tc Nc ( Spac ‘m) +c 2c) = ( Tc Nc ( Spac ‘M) +c 2c))
257	251, 256	eqeq12d 2367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (m = M → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c) ↔ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))
258	255, 257	orbi12d 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ (m = M → (( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)) ↔ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))
259	250, 258	anbi12d 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (m = M → ((( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))) ↔ (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))
260	247, 259	imbi12d 311	. . . . . . . . . . . . . . . . . 18 ⊢ (m = M → (( Nc ( Spac ‘m) = (1c +c x) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) ↔ ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
261	260	rspccv 2953	. . . . . . . . . . . . . . . . 17 ⊢ (∀m ∈ NC ( Nc ( Spac ‘m) = (1c +c x) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))) → (M ∈ NC → ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
262	244, 261	syl6com 31	. . . . . . . . . . . . . . . 16 ⊢ ( ≤c We NC → (x ∈ Nn → (M ∈ NC → ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))))
263	262	com23 72	. . . . . . . . . . . . . . 15 ⊢ ( ≤c We NC → (M ∈ NC → (x ∈ Nn → ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))))
264	263	imp 418	. . . . . . . . . . . . . 14 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (x ∈ Nn → ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
265	264	adantr 451	. . . . . . . . . . . . 13 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) → (x ∈ Nn → ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
266	265	adantr 451	. . . . . . . . . . . 12 ⊢ (((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) ∧ x ∈ NC ) → (x ∈ Nn → ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
267	27, 31, 266	3syld 51	. . . . . . . . . . 11 ⊢ (((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) ∧ x ∈ NC ) → (t = (1c +c x) → ( Nc ( Spac ‘M) = (1c +c x) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
268	267	imp3a 420	. . . . . . . . . 10 ⊢ (((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) ∧ x ∈ NC ) → ((t = (1c +c x) ∧ Nc ( Spac ‘M) = (1c +c x)) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))
269	15, 268	syl5 28	. . . . . . . . 9 ⊢ (((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) ∧ x ∈ NC ) → (( Nc ( Spac ‘M) = (1c +c x) ∧ t = Nc ( Spac ‘M)) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))
270	269	exp3a 425	. . . . . . . 8 ⊢ (((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) ∧ x ∈ NC ) → ( Nc ( Spac ‘M) = (1c +c x) → (t = Nc ( Spac ‘M) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
271	270	rexlimdva 2739	. . . . . . 7 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) → (∃x ∈ NC Nc ( Spac ‘M) = (1c +c x) → (t = Nc ( Spac ‘M) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
272	9, 271	syl5bi 208	. . . . . 6 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) → (1c ≤c Nc ( Spac ‘M) → (t = Nc ( Spac ‘M) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))))
273	4, 272	mpd 14	. . . . 5 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ t ∈ Nn ) → (t = Nc ( Spac ‘M) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))
274	273	rexlimdva 2739	. . . 4 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (∃t ∈ Nn t = Nc ( Spac ‘M) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))
275	2, 274	syl5bi 208	. . 3 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc ( Spac ‘M) ∈ Nn → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))
276	1, 275	syl5bi 208	. 2 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (( Spac ‘M) ∈ Fin → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c)))))
277	276	3impia 1148	1 ⊢ (( ≤c We NC ∧ M ∈ NC ∧ ( Spac ‘M) ∈ Fin ) → (( Spac ‘ Tc M) ∈ Fin ∧ ( Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) = ( Tc Nc ( Spac ‘M) +c 2c))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   Fin cfin 4377   class class class wbr 4640   ‘cfv 4782  (class class class)co 5526   We cwe 5896   NC cncs 6089   ≤_c clec 6090   <_c cltc 6091   Nc cnc 6092   T_c ctc 6094  2_cc2c 6095  3_cc3c 6096   ↑_c cce 6097   Sp_ac cspac 6274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-tp 3744  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  df-clos1 5874  df-trans 5900  df-antisym 5902  df-partial 5903  df-connex 5904  df-strict 5905  df-we 5907  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-lec 6100  df-ltc 6101  df-nc 6102  df-tc 6104  df-2c 6105  df-3c 6106  df-ce 6107  df-tcfn 6108  df-spac 6275

This theorem is referenced by:  nchoicelem19  6308  nchoice  6309