fzisoeu - Mathbox for Glauco Siliprandi

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Theorem fzisoeu 44000

Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 14422 for the base index and also states the uniqueness condition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fzisoeu.h	⊢ (𝜑 → 𝐻 ∈ Fin)
fzisoeu.or	⊢ (𝜑 → < Or 𝐻)
fzisoeu.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
fzisoeu.4	⊢ 𝑁 = ((♯‘𝐻) + (𝑀 − 1))

**Assertion**
Ref	Expression
fzisoeu	⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))

Distinct variable groups: 𝑓,𝐻 𝑓,𝑀 𝑓,𝑁 Allowed substitution hint: 𝜑(𝑓)

Proof of Theorem fzisoeu Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzssz 13502	. . . . . . . . 9 ⊢ (𝑀...𝑁) ⊆ ℤ
2		zssre 12564	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
3	1, 2	sstri 3991	. . . . . . . 8 ⊢ (𝑀...𝑁) ⊆ ℝ
4		ltso 11293	. . . . . . . 8 ⊢ < Or ℝ
5		soss 5608	. . . . . . . 8 ⊢ ((𝑀...𝑁) ⊆ ℝ → ( < Or ℝ → < Or (𝑀...𝑁)))
6	3, 4, 5	mp2 9	. . . . . . 7 ⊢ < Or (𝑀...𝑁)
7		fzfi 13936	. . . . . . 7 ⊢ (𝑀...𝑁) ∈ Fin
8		fz1iso 14422	. . . . . . 7 ⊢ (( < Or (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → ∃ℎ ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)))
9	6, 7, 8	mp2an 690	. . . . . 6 ⊢ ∃ℎ ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁))
10		fzisoeu.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 = ((♯‘𝐻) + (𝑀 − 1))
11		fveq2 6891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐻 = ∅ → (♯‘𝐻) = (♯‘∅))
12		hash0 14326	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘∅) = 0
13	11, 12	eqtrdi 2788	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 = ∅ → (♯‘𝐻) = 0)
14	13	oveq1d 7423	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 = ∅ → ((♯‘𝐻) + (𝑀 − 1)) = (0 + (𝑀 − 1)))
15	10, 14	eqtrid 2784	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ → 𝑁 = (0 + (𝑀 − 1)))
16	15	oveq2d 7424	. . . . . . . . . . . . . 14 ⊢ (𝐻 = ∅ → (𝑀...𝑁) = (𝑀...(0 + (𝑀 − 1))))
17	16	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = (𝑀...(0 + (𝑀 − 1))))
18		fzisoeu.m	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	zcnd 12666	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℂ)
20		1cnd 11208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℂ)
21	19, 20	subcld 11570	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℂ)
22	21	addlidd 11414	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + (𝑀 − 1)) = (𝑀 − 1))
23	22	oveq2d 7424	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = (𝑀...(𝑀 − 1)))
24	18	zred 12665	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	24	ltm1d 12145	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
26		peano2zm 12604	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
27	18, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
28		fzn 13516	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
29	18, 27, 28	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
30	25, 29	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
31	23, 30	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = ∅)
32	31	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...(0 + (𝑀 − 1))) = ∅)
33		eqcom 2739	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ ↔ ∅ = 𝐻)
34	33	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (𝐻 = ∅ → ∅ = 𝐻)
35	34	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → ∅ = 𝐻)
36	17, 32, 35	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = 𝐻)
37	36	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
38	20, 19	pncan3d 11573	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 + (𝑀 − 1)) = 𝑀)
39	38	eqcomd 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 = (1 + (𝑀 − 1)))
40	39	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 = (1 + (𝑀 − 1)))
41		1red 11214	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ∈ ℝ)
42		neqne 2948	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝐻 = ∅ → 𝐻 ≠ ∅)
43	42	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝐻 ≠ ∅)
44		fzisoeu.h	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐻 ∈ Fin)
45	44	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝐻 ∈ Fin)
46		hashnncl 14325	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 ∈ Fin → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
47	45, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
48	43, 47	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘𝐻) ∈ ℕ)
49	48	nnred 12226	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘𝐻) ∈ ℝ)
50	27	zred 12665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
51	50	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑀 − 1) ∈ ℝ)
52	48	nnge1d 12259	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ≤ (♯‘𝐻))
53	41, 49, 51, 52	leadd1dd 11827	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ ((♯‘𝐻) + (𝑀 − 1)))
54	53, 10	breqtrrdi 5190	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ 𝑁)
55	40, 54	eqbrtrd 5170	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ≤ 𝑁)
56	18	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ∈ ℤ)
57		hashcl 14315	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 ∈ Fin → (♯‘𝐻) ∈ ℕ₀)
58		nn0z 12582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐻) ∈ ℕ₀ → (♯‘𝐻) ∈ ℤ)
59	44, 57, 58	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝐻) ∈ ℤ)
60	59, 27	zaddcld 12669	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝐻) + (𝑀 − 1)) ∈ ℤ)
61	10, 60	eqeltrid 2837	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ ℤ)
63		eluz 12835	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
64	56, 62, 63	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑁 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
65	55, 64	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ (ℤ_≥‘𝑀))
66		hashfz 14386	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (♯‘(𝑀...𝑁)) = ((𝑁 − 𝑀) + 1))
67	65, 66	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = ((𝑁 − 𝑀) + 1))
68	10	oveq1i 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 − 𝑀) = (((♯‘𝐻) + (𝑀 − 1)) − 𝑀)
69	44, 57	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝐻) ∈ ℕ₀)
70	69	nn0cnd 12533	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝐻) ∈ ℂ)
71	70, 21, 19	addsubassd 11590	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((♯‘𝐻) + (𝑀 − 1)) − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)))
72	68, 71	eqtrid 2784	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)))
73	20	negcld 11557	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → -1 ∈ ℂ)
74	19, 20	negsubd 11576	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + -1) = (𝑀 − 1))
75	74	eqcomd 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) = (𝑀 + -1))
76	19, 73, 75	mvrladdd 11626	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = -1)
77	76	oveq2d 7424	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)) = ((♯‘𝐻) + -1))
78	72, 77	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + -1))
79	78	oveq1d 7423	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) = (((♯‘𝐻) + -1) + 1))
80	79	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((𝑁 − 𝑀) + 1) = (((♯‘𝐻) + -1) + 1))
81	70, 20	negsubd 11576	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐻) + -1) = ((♯‘𝐻) − 1))
82	81	oveq1d 7423	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) = (((♯‘𝐻) − 1) + 1))
83	70, 20	npcand 11574	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝐻) − 1) + 1) = (♯‘𝐻))
84	82, 83	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) = (♯‘𝐻))
85	84	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (((♯‘𝐻) + -1) + 1) = (♯‘𝐻))
86	67, 80, 85	3eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
87	37, 86	pm2.61dan 811	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
88	87	oveq2d 7424	. . . . . . . . 9 ⊢ (𝜑 → (1...(♯‘(𝑀...𝑁))) = (1...(♯‘𝐻)))
89		isoeq4 7316	. . . . . . . . 9 ⊢ ((1...(♯‘(𝑀...𝑁))) = (1...(♯‘𝐻)) → (ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) ↔ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁))))
90	88, 89	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) ↔ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁))))
91	90	biimpd 228	. . . . . . 7 ⊢ (𝜑 → (ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) → ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁))))
92	91	eximdv 1920	. . . . . 6 ⊢ (𝜑 → (∃ℎ ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) → ∃ℎ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁))))
93	9, 92	mpi 20	. . . . 5 ⊢ (𝜑 → ∃ℎ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)))
94		fzisoeu.or	. . . . . 6 ⊢ (𝜑 → < Or 𝐻)
95		fz1iso 14422	. . . . . 6 ⊢ (( < Or 𝐻 ∧ 𝐻 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))
96	94, 44, 95	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))
97		exdistrv 1959	. . . . 5 ⊢ (∃ℎ∃𝑔(ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) ↔ (∃ℎ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)))
98	93, 96, 97	sylanbrc 583	. . . 4 ⊢ (𝜑 → ∃ℎ∃𝑔(ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)))
99		isocnv 7326	. . . . . . . 8 ⊢ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻))))
100	99	ad2antrl 726	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻))))
101		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))
102		isotr 7332	. . . . . . 7 ⊢ ((^◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻))) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
103	100, 101, 102	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
104	103	ex 413	. . . . 5 ⊢ (𝜑 → ((ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
105	104	2eximdv 1922	. . . 4 ⊢ (𝜑 → (∃ℎ∃𝑔(ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) → ∃ℎ∃𝑔(𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
106	98, 105	mpd 15	. . 3 ⊢ (𝜑 → ∃ℎ∃𝑔(𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
107		vex 3478	. . . . . . 7 ⊢ 𝑔 ∈ V
108		vex 3478	. . . . . . . 8 ⊢ ℎ ∈ V
109	108	cnvex 7915	. . . . . . 7 ⊢ ^◡ℎ ∈ V
110	107, 109	coex 7920	. . . . . 6 ⊢ (𝑔 ∘ ^◡ℎ) ∈ V
111		isoeq1 7313	. . . . . 6 ⊢ (𝑓 = (𝑔 ∘ ^◡ℎ) → (𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
112	110, 111	spcev 3596	. . . . 5 ⊢ ((𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
113	112	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)))
114	113	exlimdvv 1937	. . 3 ⊢ (𝜑 → (∃ℎ∃𝑔(𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)))
115	106, 114	mpd 15	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
116		ltwefz 13927	. . 3 ⊢ < We (𝑀...𝑁)
117		wemoiso 7959	. . 3 ⊢ ( < We (𝑀...𝑁) → ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
118	116, 117	mp1i 13	. 2 ⊢ (𝜑 → ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
119		df-eu 2563	. 2 ⊢ (∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ∧ ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)))
120	115, 118, 119	sylanbrc 583	1 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃*wmo 2532 ∃!weu 2562 ≠ wne 2940 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 Or wor 5587 We wwe 5630 ^◡ccnv 5675 ∘ ccom 5680 ‘cfv 6543 Isom wiso 6544 (class class class)co 7408 Fincfn 8938 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 < clt 11247 ≤ cle 11248 − cmin 11443 -cneg 11444 ℕcn 12211 ℕ₀cn0 12471 ℤcz 12557 ℤ_≥cuz 12821 ...cfz 13483 ♯chash 14289

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-hash 14290

This theorem is referenced by: fourierdlem36 44849

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