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Theorem chnccat 18681

Description: Concatenate two chains. (Contributed by Ender Ting, 20-Jan-2026.)

**Hypotheses**
Ref	Expression
chnccat.1	⊢ (𝜑 → 𝑇 ∈ ( < Chain 𝐴))
chnccat.2	⊢ (𝜑 → 𝑈 ∈ ( < Chain 𝐴))
chnccat.3	⊢ (𝜑 → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0)))

**Assertion**
Ref	Expression
chnccat	⊢ (𝜑 → (𝑇 ++ 𝑈) ∈ ( < Chain 𝐴))

Proof of Theorem chnccat Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chnccat.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ( < Chain 𝐴))
2	1	chnwrd 18663	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
3		chnccat.2	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ( < Chain 𝐴))
4	3	chnwrd 18663	. . 3 ⊢ (𝜑 → 𝑈 ∈ Word 𝐴)
5		ccatcl 14610	. . 3 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴) → (𝑇 ++ 𝑈) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → (𝑇 ++ 𝑈) ∈ Word 𝐴)
7		eqidd 2762	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) = (♯‘𝑇))
8	7, 2	wrdfd 14555	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
9	8	fdmd 6716	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 = (0..^(♯‘𝑇)))
10	9	difeq1d 4079	. . . . . . . . 9 ⊢ (𝜑 → (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
11	10	eleq2d 2847	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
12	11	biimpar 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
13		snsspr1 4779	. . . . . . . . . 10 ⊢ {0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))}
14		sscon 4096	. . . . . . . . . 10 ⊢ ({0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} → (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ (dom 𝑇 ∖ {0}))
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ (dom 𝑇 ∖ {0})
16	15	sseli 3932	. . . . . . . 8 ⊢ (𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ (dom 𝑇 ∖ {0}))
17		ischn 18662	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ( < Chain 𝐴) ↔ (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
18	1, 17	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
19	18	simprd 500	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
20	19	r19.21bi 3255	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom 𝑇 ∖ {0})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
21	16, 20	sylan2 604	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
22	12, 21	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
23	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑇 ∈ Word 𝐴)
24	4	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ Word 𝐴)
25		sscon 4096	. . . . . . . . . . 11 ⊢ ({0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} → ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ((0..^(♯‘𝑇)) ∖ {0}))
26	13, 25	ax-mp 5	. . . . . . . . . 10 ⊢ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ((0..^(♯‘𝑇)) ∖ {0})
27	26	sseli 3932	. . . . . . . . 9 ⊢ (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}))
28	27	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}))
29		lencl 14569	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
30	2, 29	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
31	30	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℕ₀)
32	28, 31	elfzodif0 13798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
33		ccatval1 14613	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
34	23, 24, 32, 33	syl3anc 1396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
35		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
36	35	eldifad 3916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ (0..^(♯‘𝑇)))
37		ccatval1 14613	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑇‘𝑛))
38	23, 24, 36, 37	syl3anc 1396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑇‘𝑛))
39	22, 34, 38	3brtr4d 5142	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
40	39	adantlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
41		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
42	41	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
43		noel 4290	. . . . . . . 8 ⊢ ¬ 𝑛 ∈ ∅
44		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑇 = ∅ → (♯‘𝑇) = (♯‘∅))
45		hash0 14402	. . . . . . . . . . . . . 14 ⊢ (♯‘∅) = 0
46	44, 45	eqtrdi 2812	. . . . . . . . . . . . 13 ⊢ (𝑇 = ∅ → (♯‘𝑇) = 0)
47	46	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → (♯‘𝑇) = 0)
48	47	sneqd 4600	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → {(♯‘𝑇)} = {0})
49	48	difeq1d 4079	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ({0} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
50		difpr 4770	. . . . . . . . . . 11 ⊢ ({0} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = (({0} ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))})
51		difid 4331	. . . . . . . . . . . 12 ⊢ ({0} ∖ {0}) = ∅
52	51	difeq1i 4076	. . . . . . . . . . 11 ⊢ (({0} ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}) = (∅ ∖ {((♯‘𝑇) + (♯‘𝑈))})
53		0dif 4362	. . . . . . . . . . 11 ⊢ (∅ ∖ {((♯‘𝑇) + (♯‘𝑈))}) = ∅
54	50, 52, 53	3eqtri 2788	. . . . . . . . . 10 ⊢ ({0} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅
55	49, 54	eqtrdi 2812	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅)
56	55	eleq2d 2847	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ 𝑛 ∈ ∅))
57	43, 56	mtbiri 330	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
58	42, 57	pm2.21dd 198	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
59		eldifi 4084	. . . . . . . . 9 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ {(♯‘𝑇)})
60	59	elsnd 4606	. . . . . . . 8 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 = (♯‘𝑇))
61	60	ad2antlr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → 𝑛 = (♯‘𝑇))
62		vex 3457	. . . . . . . . 9 ⊢ 𝑛 ∈ V
63	62	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → 𝑛 ∈ V)
64		eldifn 4085	. . . . . . . . . 10 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
65	64	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
66		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑈 = ∅ → (♯‘𝑈) = (♯‘∅))
67	66, 45	eqtrdi 2812	. . . . . . . . . . . . 13 ⊢ (𝑈 = ∅ → (♯‘𝑈) = 0)
68	67	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → (♯‘𝑈) = 0)
69	68	oveq2d 7426	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((♯‘𝑇) + (♯‘𝑈)) = ((♯‘𝑇) + 0))
70		nn0cn 12513	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℕ₀ → (♯‘𝑇) ∈ ℂ)
71	2, 29, 70	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑇) ∈ ℂ)
72	71	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℂ)
73	72	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → (♯‘𝑇) ∈ ℂ)
74	73	addridd 11409	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((♯‘𝑇) + 0) = (♯‘𝑇))
75	69, 74	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((♯‘𝑇) + (♯‘𝑈)) = (♯‘𝑇))
76	75	preq2d 4705	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → {0, ((♯‘𝑇) + (♯‘𝑈))} = {0, (♯‘𝑇)})
77	65, 76	neleqtrd 2883	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ¬ 𝑛 ∈ {0, (♯‘𝑇)})
78	63, 77	nelpr2 4618	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → 𝑛 ≠ (♯‘𝑇))
79	61, 78	pm2.21ddne 3040	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
80		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (lastS‘𝑇) < (𝑈‘0))
81	2	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑇 ∈ Word 𝐴)
82	81	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 𝑇 ∈ Word 𝐴)
83	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ Word 𝐴)
84	83	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 𝑈 ∈ Word 𝐴)
85	41	eldifad 3916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ {(♯‘𝑇)})
86	85	elsnd 4606	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 = (♯‘𝑇))
87	86	oveq1d 7425	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) = ((♯‘𝑇) − 1))
88	81, 29	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℕ₀)
89		sscon 4096	. . . . . . . . . . . . . . . . . . 19 ⊢ ({0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ({(♯‘𝑇)} ∖ {0}))
90	13, 89	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ({(♯‘𝑇)} ∖ {0})
91	90	sseli 3932	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
92	60, 91	eqeltrrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → (♯‘𝑇) ∈ ({(♯‘𝑇)} ∖ {0}))
93	92	eldifbd 3917	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → ¬ (♯‘𝑇) ∈ {0})
94	93	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ¬ (♯‘𝑇) ∈ {0})
95	88, 94	eldifd 3915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ (ℕ₀ ∖ {0}))
96		dfn2 12516	. . . . . . . . . . . . 13 ⊢ ℕ = (ℕ₀ ∖ {0})
97	95, 96	eleqtrrdi 2872	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℕ)
98		fzo0end 13786	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
99	97, 98	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
100	87, 99	eqeltrd 2861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
101	100	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
102	82, 84, 101, 33	syl3anc 1396	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
103	60	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → (𝑛 − 1) = ((♯‘𝑇) − 1))
104	103	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) = ((♯‘𝑇) − 1))
105	104	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) = (𝑇‘((♯‘𝑇) − 1)))
106		lsw 14600	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
107	81, 106	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
108	105, 107	eqtr4d 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) = (lastS‘𝑇))
109	108	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (𝑇‘(𝑛 − 1)) = (lastS‘𝑇))
110	102, 109	eqtrd 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (lastS‘𝑇))
111	86	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 𝑛 = (♯‘𝑇))
112	111	fveq2d 6885	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘𝑛) = ((𝑇 ++ 𝑈)‘(♯‘𝑇)))
113		lencl 14569	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ Word 𝐴 → (♯‘𝑈) ∈ ℕ₀)
114	4, 113	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑈) ∈ ℕ₀)
115	114	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ∈ ℕ₀)
116	81	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → 𝑇 ∈ Word 𝐴)
117	116, 29, 70	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → (♯‘𝑇) ∈ ℂ)
118		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → (♯‘𝑈) = 0)
119	117, 118	jca 520	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0))
120		prid2g 4726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑇) ∈ ℂ → (♯‘𝑇) ∈ {0, (♯‘𝑇)})
121	120	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → (♯‘𝑇) ∈ {0, (♯‘𝑇)})
122		simpr 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → (♯‘𝑈) = 0)
123	122	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) + (♯‘𝑈)) = ((♯‘𝑇) + 0))
124		addrid 11389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑇) ∈ ℂ → ((♯‘𝑇) + 0) = (♯‘𝑇))
125	124	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) + 0) = (♯‘𝑇))
126	123, 125	eqtrd 2796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) + (♯‘𝑈)) = (♯‘𝑇))
127	126	preq2d 4705	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → {0, ((♯‘𝑇) + (♯‘𝑈))} = {0, (♯‘𝑇)})
128	121, 127	eleqtrrd 2864	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → (♯‘𝑇) ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
129	128	snssd 4751	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → {(♯‘𝑇)} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))})
130		ssdif0 4320	. . . . . . . . . . . . . . . . 17 ⊢ ({(♯‘𝑇)} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} ↔ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅)
131	129, 130	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅)
132		nel02 4291	. . . . . . . . . . . . . . . 16 ⊢ (({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅ → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
133	119, 131, 132	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
134	133	ex 417	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((♯‘𝑈) = 0 → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
135	41, 134	mt2d 137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ¬ (♯‘𝑈) = 0)
136	135	neqned 2963	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ≠ 0)
137		elnnne0 12517	. . . . . . . . . . . 12 ⊢ ((♯‘𝑈) ∈ ℕ ↔ ((♯‘𝑈) ∈ ℕ₀ ∧ (♯‘𝑈) ≠ 0))
138	115, 136, 137	sylanbrc 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ∈ ℕ)
139	138	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (♯‘𝑈) ∈ ℕ)
140		lbfzo0 13727	. . . . . . . . . 10 ⊢ (0 ∈ (0..^(♯‘𝑈)) ↔ (♯‘𝑈) ∈ ℕ)
141	139, 140	sylibr 237	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 0 ∈ (0..^(♯‘𝑈)))
142		addlid 11392	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℂ → (0 + (♯‘𝑇)) = (♯‘𝑇))
143	142	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑇) ∈ ℂ → (♯‘𝑇) = (0 + (♯‘𝑇)))
144	143	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℂ → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))))
145	29, 70, 144	3syl 19	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))))
146	145	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 0 ∈ (0..^(♯‘𝑈))) → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))))
147		ccatval3 14615	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 0 ∈ (0..^(♯‘𝑈))) → ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))) = (𝑈‘0))
148	146, 147	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 0 ∈ (0..^(♯‘𝑈))) → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = (𝑈‘0))
149	82, 84, 141, 148	syl3anc 1396	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = (𝑈‘0))
150	112, 149	eqtrd 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑈‘0))
151	80, 110, 150	3brtr4d 5142	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
152		chnccat.3	. . . . . . 7 ⊢ (𝜑 → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0)))
153	152	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0)))
154	58, 79, 151, 153	mpjao3dan 1457	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
155	154	adantlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
156		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
157		eldifi 4084	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}))
158	157	eldifad 3916	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
159		elfzoelz 13686	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) → 𝑛 ∈ ℤ)
160	158, 159	syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ℤ)
161		zcn 12595	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
162	156, 160, 161	3syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ℂ)
163		1cnd 11201	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 1 ∈ ℂ)
164	71	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℂ)
165	162, 163, 164	sub32d 11600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑛 − 1) − (♯‘𝑇)) = ((𝑛 − (♯‘𝑇)) − 1))
166	165	fveq2d 6885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑈‘((𝑛 − 1) − (♯‘𝑇))) = (𝑈‘((𝑛 − (♯‘𝑇)) − 1)))
167	3	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ ( < Chain 𝐴))
168	158	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
169		nn0z 12614	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑈) ∈ ℕ₀ → (♯‘𝑈) ∈ ℤ)
170	4, 113, 169	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑈) ∈ ℤ)
171	170	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ∈ ℤ)
172		fzosubel3 13754	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑈) ∈ ℤ) → (𝑛 − (♯‘𝑇)) ∈ (0..^(♯‘𝑈)))
173	168, 171, 172	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − (♯‘𝑇)) ∈ (0..^(♯‘𝑈)))
174		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝜑)
175	156	eldifad 3916	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}))
176	174, 175	jca 520	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})))
177		eldifi 4084	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
178	177, 159, 161	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ∈ ℂ)
179	178	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ∈ ℂ)
180	71	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ∈ ℂ)
181		eldifsni 4757	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ≠ (♯‘𝑇))
182	181	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ≠ (♯‘𝑇))
183	179, 180, 182	subne0d 11577	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − (♯‘𝑇)) ≠ 0)
184		nelsn 4631	. . . . . . . . . . 11 ⊢ ((𝑛 − (♯‘𝑇)) ≠ 0 → ¬ (𝑛 − (♯‘𝑇)) ∈ {0})
185	176, 183, 184	3syl 19	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ¬ (𝑛 − (♯‘𝑇)) ∈ {0})
186	173, 185	eldifd 3915	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − (♯‘𝑇)) ∈ ((0..^(♯‘𝑈)) ∖ {0}))
187		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑈) = (♯‘𝑈))
188	187, 4	wrdfd 14555	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈:(0..^(♯‘𝑈))⟶𝐴)
189	188	fdmd 6716	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑈 = (0..^(♯‘𝑈)))
190	189	difeq1d 4079	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑈 ∖ {0}) = ((0..^(♯‘𝑈)) ∖ {0}))
191	190	eleq2d 2847	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑛 − (♯‘𝑇)) ∈ (dom 𝑈 ∖ {0}) ↔ (𝑛 − (♯‘𝑇)) ∈ ((0..^(♯‘𝑈)) ∖ {0})))
192	191	biimpar 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 − (♯‘𝑇)) ∈ ((0..^(♯‘𝑈)) ∖ {0})) → (𝑛 − (♯‘𝑇)) ∈ (dom 𝑈 ∖ {0}))
193	186, 192	syldan 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − (♯‘𝑇)) ∈ (dom 𝑈 ∖ {0}))
194	167, 193	chnltm1 18664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑈‘((𝑛 − (♯‘𝑇)) − 1)) < (𝑈‘(𝑛 − (♯‘𝑇))))
195	166, 194	eqbrtrd 5132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑈‘((𝑛 − 1) − (♯‘𝑇))) < (𝑈‘(𝑛 − (♯‘𝑇))))
196	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑇 ∈ Word 𝐴)
197	4	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ Word 𝐴)
198	177, 159	syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ∈ ℤ)
199	198	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ∈ ℤ)
200		nn0z 12614	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑇) ∈ ℕ₀ → (♯‘𝑇) ∈ ℤ)
201	2, 29, 200	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) ∈ ℤ)
202	201	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ∈ ℤ)
203	199, 202	jca 520	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ))
204		elfzole1 13695	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) → (♯‘𝑇) ≤ 𝑛)
205	177, 204	syl 18	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → (♯‘𝑇) ≤ 𝑛)
206	205	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ≤ 𝑛)
207		eldifn 4085	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → ¬ 𝑛 ∈ {(♯‘𝑇)})
208		velsn 4604	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {(♯‘𝑇)} ↔ 𝑛 = (♯‘𝑇))
209	208	biimpri 231	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (♯‘𝑇) → 𝑛 ∈ {(♯‘𝑇)})
210	209	necon3bi 2982	. . . . . . . . . . . . 13 ⊢ (¬ 𝑛 ∈ {(♯‘𝑇)} → 𝑛 ≠ (♯‘𝑇))
211	210	necomd 3011	. . . . . . . . . . . 12 ⊢ (¬ 𝑛 ∈ {(♯‘𝑇)} → (♯‘𝑇) ≠ 𝑛)
212	207, 211	syl 18	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → (♯‘𝑇) ≠ 𝑛)
213	212	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ≠ 𝑛)
214		simp1r 1215	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ∈ ℤ)
215	214	zred 12699	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ∈ ℝ)
216		simp1l 1214	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 𝑛 ∈ ℤ)
217	216	zred 12699	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 𝑛 ∈ ℝ)
218		simp2 1153	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ≤ 𝑛)
219		simp3 1154	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ≠ 𝑛)
220	219	necomd 3011	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 𝑛 ≠ (♯‘𝑇))
221	215, 217, 218, 220	leneltd 11363	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) < 𝑛)
222		simp1 1152	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ))
223	222	ancomd 466	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) ∈ ℤ ∧ 𝑛 ∈ ℤ))
224		zltp1le 12643	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑇) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((♯‘𝑇) < 𝑛 ↔ ((♯‘𝑇) + 1) ≤ 𝑛))
225	223, 224	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) < 𝑛 ↔ ((♯‘𝑇) + 1) ≤ 𝑛))
226	221, 225	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) + 1) ≤ 𝑛)
227		peano2re 11382	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℝ → ((♯‘𝑇) + 1) ∈ ℝ)
228	215, 227	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) + 1) ∈ ℝ)
229		1red 11208	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 1 ∈ ℝ)
230	228, 217, 229	lesub1d 11820	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (((♯‘𝑇) + 1) ≤ 𝑛 ↔ (((♯‘𝑇) + 1) − 1) ≤ (𝑛 − 1)))
231		zcn 12595	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑇) ∈ ℤ → (♯‘𝑇) ∈ ℂ)
232		1cnd 11201	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑇) ∈ ℤ → 1 ∈ ℂ)
233	231, 232	pncand 11569	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℤ → (((♯‘𝑇) + 1) − 1) = (♯‘𝑇))
234	233	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) → (((♯‘𝑇) + 1) − 1) = (♯‘𝑇))
235	234	3ad2ant1 1149	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (((♯‘𝑇) + 1) − 1) = (♯‘𝑇))
236	235	breq1d 5118	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((((♯‘𝑇) + 1) − 1) ≤ (𝑛 − 1) ↔ (♯‘𝑇) ≤ (𝑛 − 1)))
237	230, 236	bitrd 282	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (((♯‘𝑇) + 1) ≤ 𝑛 ↔ (♯‘𝑇) ≤ (𝑛 − 1)))
238	226, 237	mpbid 235	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ≤ (𝑛 − 1))
239	203, 206, 213, 238	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ≤ (𝑛 − 1))
240	199	zred 12699	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ∈ ℝ)
241		peano2rem 11524	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
242	240, 241	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) ∈ ℝ)
243	201, 170	zaddcld 12703	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)
244	243	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)
245	244	zred 12699	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℝ)
246	240	ltm1d 12146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) < 𝑛)
247		elfzolt2 13696	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) → 𝑛 < ((♯‘𝑇) + (♯‘𝑈)))
248	177, 247	syl 18	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 < ((♯‘𝑇) + (♯‘𝑈)))
249	248	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 < ((♯‘𝑇) + (♯‘𝑈)))
250	242, 240, 245, 246, 249	lttrd 11370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) < ((♯‘𝑇) + (♯‘𝑈)))
251		peano2zm 12636	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
252	199, 251	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) ∈ ℤ)
253		elfzo 13688	. . . . . . . . . 10 ⊢ (((𝑛 − 1) ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ) → ((𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ↔ ((♯‘𝑇) ≤ (𝑛 − 1) ∧ (𝑛 − 1) < ((♯‘𝑇) + (♯‘𝑈)))))
254	252, 202, 244, 253	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → ((𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ↔ ((♯‘𝑇) ≤ (𝑛 − 1) ∧ (𝑛 − 1) < ((♯‘𝑇) + (♯‘𝑈)))))
255	239, 250, 254	mpbir2and 725	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
256	157, 255	sylan2 604	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
257		ccatval2 14614	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑈‘((𝑛 − 1) − (♯‘𝑇))))
258	196, 197, 256, 257	syl3anc 1396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑈‘((𝑛 − 1) − (♯‘𝑇))))
259		ccatval2 14614	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑈‘(𝑛 − (♯‘𝑇))))
260	196, 197, 168, 259	syl3anc 1396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑈‘(𝑛 − (♯‘𝑇))))
261	195, 258, 260	3brtr4d 5142	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
262	261	adantlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
263		ccatlen 14611	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴) → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
264	2, 4, 263	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
265	264	eqcomd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑇) + (♯‘𝑈)) = (♯‘(𝑇 ++ 𝑈)))
266	265, 6	wrdfd 14555	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ++ 𝑈):(0..^((♯‘𝑇) + (♯‘𝑈)))⟶𝐴)
267	266	fdmd 6716	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 ++ 𝑈) = (0..^((♯‘𝑇) + (♯‘𝑈))))
268	267	difeq1d 4079	. . . . . . . . 9 ⊢ (𝜑 → (dom (𝑇 ++ 𝑈) ∖ {0}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
269		fzonel 13701	. . . . . . . . . . . 12 ⊢ ¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ (0..^((♯‘𝑇) + (♯‘𝑈)))
270		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
271	270	eldifad 3916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ (0..^((♯‘𝑇) + (♯‘𝑈))))
272	271	ex 417	. . . . . . . . . . . 12 ⊢ (𝜑 → (((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) → ((♯‘𝑇) + (♯‘𝑈)) ∈ (0..^((♯‘𝑇) + (♯‘𝑈)))))
273	269, 272	mtoi 202	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
274		difsn 4765	. . . . . . . . . . . 12 ⊢ (¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) → (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
275	274	eqcomd 2767	. . . . . . . . . . 11 ⊢ (¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) → ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) = (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}))
276	273, 275	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) = (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}))
277		difpr 4770	. . . . . . . . . 10 ⊢ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))})
278	276, 277	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝜑 → ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
279	268, 278	eqtrd 2796	. . . . . . . 8 ⊢ (𝜑 → (dom (𝑇 ++ 𝑈) ∖ {0}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
280	279	eleq2d 2847	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0}) ↔ 𝑛 ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
281		eldif 3914	. . . . . . 7 ⊢ (𝑛 ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))}))
282	280, 281	bitrdi 290	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0}) ↔ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})))
283		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → 𝑛 ∈ (0..^(♯‘𝑇)))
284		simplrr 789	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
285	283, 284	eldifd 3915	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
286	285	orcd 886	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
287		exmidd 908	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} ∨ ¬ 𝑛 ∈ {(♯‘𝑇)}))
288		idd 25	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ {(♯‘𝑇)}))
289		simplrr 789	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
290	288, 289	jctird 535	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} → (𝑛 ∈ {(♯‘𝑇)} ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})))
291		eldif 3914	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ (𝑛 ∈ {(♯‘𝑇)} ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))}))
292	290, 291	imbitrrdi 255	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
293		idd 25	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → ¬ 𝑛 ∈ {(♯‘𝑇)}))
294		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
295	293, 294	jctild 534	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {(♯‘𝑇)})))
296		eldif 3914	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ↔ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {(♯‘𝑇)}))
297	295, 296	imbitrrdi 255	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})))
298	297, 289	jctird 535	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})))
299		eldif 3914	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))}))
300	298, 299	imbitrrdi 255	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
301	292, 300	orim12d 979	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ((𝑛 ∈ {(♯‘𝑇)} ∨ ¬ 𝑛 ∈ {(♯‘𝑇)}) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
302	287, 301	mpd 16	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
303	302	olcd 887	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
304	201	anim1ci 627	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑇) ∈ ℤ))
305	304	adantrr 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑇) ∈ ℤ))
306		fzospliti 13719	. . . . . . . 8 ⊢ ((𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑛 ∈ (0..^(♯‘𝑇)) ∨ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))))
307	305, 306	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 ∈ (0..^(♯‘𝑇)) ∨ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))))
308	286, 303, 307	mpjaodan 973	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
309	282, 308	sylbida 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
310		3orass 1104	. . . . 5 ⊢ ((𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ↔ (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
311	309, 310	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
312	40, 155, 262, 311	mpjao3dan 1457	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
313	312	ralrimiva 3155	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
314		ischn 18662	. 2 ⊢ ((𝑇 ++ 𝑈) ∈ ( < Chain 𝐴) ↔ ((𝑇 ++ 𝑈) ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛)))
315	6, 313, 314	sylanbrc 594	1 ⊢ (𝜑 → (𝑇 ++ 𝑈) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ∀wral 3077 Vcvv 3453 ∖ cdif 3901 ⊆ wss 3904 ∅c0 4285 {csn 4588 {cpr 4590 class class class wbr 5108 dom cdm 5661 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 < clt 11242 ≤ cle 11243 − cmin 11440 ℕcn 12232 ℕ₀cn0 12503 ℤcz 12590 ..^cfzo 13681 ♯chash 14365 Word cword 14549 lastSclsw 14598 ++ cconcat 14606 Chain cchn 18660

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13682 df-hash 14366 df-word 14550 df-lsw 14599 df-concat 14607 df-chn 18661

This theorem is referenced by: (None)

W3C validator