fzisoeu - Mathbox for Glauco Siliprandi

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Theorem fzisoeu 38455

Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13103 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 12214	. . . . . . . . 9 ⊢ (𝑀...𝑁) ⊆ ℤ
2		zssre 11261	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
3	1, 2	sstri 3577	. . . . . . . 8 ⊢ (𝑀...𝑁) ⊆ ℝ
4		ltso 9997	. . . . . . . 8 ⊢ < Or ℝ
5		soss 4977	. . . . . . . 8 ⊢ ((𝑀...𝑁) ⊆ ℝ → ( < Or ℝ → < Or (𝑀...𝑁)))
6	3, 4, 5	mp2 9	. . . . . . 7 ⊢ < Or (𝑀...𝑁)
7		fzfi 12633	. . . . . . 7 ⊢ (𝑀...𝑁) ∈ Fin
8		fz1iso 13103	. . . . . . 7 ⊢ (( < Or (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → ∃ℎ ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁)))
9	6, 7, 8	mp2an 704	. . . . . 6 ⊢ ∃ℎ ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁))
10		fzisoeu.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 = ((#‘𝐻) + (𝑀 − 1))
11		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐻 = ∅ → (#‘𝐻) = (#‘∅))
12		hash0 13019	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘∅) = 0
13	11, 12	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 = ∅ → (#‘𝐻) = 0)
14	13	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 = ∅ → ((#‘𝐻) + (𝑀 − 1)) = (0 + (𝑀 − 1)))
15	10, 14	syl5eq 2656	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ → 𝑁 = (0 + (𝑀 − 1)))
16	15	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (𝐻 = ∅ → (𝑀...𝑁) = (𝑀...(0 + (𝑀 − 1))))
17	16	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = (𝑀...(0 + (𝑀 − 1))))
18		fzisoeu.m	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	zcnd 11359	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℂ)
20		1cnd 9935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℂ)
21	19, 20	subcld 10271	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℂ)
22	21	addid2d 10116	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + (𝑀 − 1)) = (𝑀 − 1))
23	22	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = (𝑀...(𝑀 − 1)))
24	18	zred 11358	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	24	ltm1d 10835	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
26		peano2zm 11297	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
27	18, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
28		fzn 12228	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
29	18, 27, 28	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
30	25, 29	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
31	23, 30	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = ∅)
32	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...(0 + (𝑀 − 1))) = ∅)
33		eqcom 2617	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ ↔ ∅ = 𝐻)
34	33	biimpi 205	. . . . . . . . . . . . . 14 ⊢ (𝐻 = ∅ → ∅ = 𝐻)
35	34	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → ∅ = 𝐻)
36	17, 32, 35	3eqtrd 2648	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = 𝐻)
37	36	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (#‘(𝑀...𝑁)) = (#‘𝐻))
38	20, 19	pncan3d 10274	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 + (𝑀 − 1)) = 𝑀)
39	38	eqcomd 2616	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 = (1 + (𝑀 − 1)))
40	39	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 = (1 + (𝑀 − 1)))
41		1red 9934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ∈ ℝ)
42		neqne 2790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝐻 = ∅ → 𝐻 ≠ ∅)
43	42	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝐻 ≠ ∅)
44		fzisoeu.h	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐻 ∈ Fin)
45	44	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝐻 ∈ Fin)
46		hashnncl 13018	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 ∈ Fin → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
47	45, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
48	43, 47	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (#‘𝐻) ∈ ℕ)
49	48	nnred 10912	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (#‘𝐻) ∈ ℝ)
50	27	zred 11358	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
51	50	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑀 − 1) ∈ ℝ)
52	48	nnge1d 10940	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ≤ (#‘𝐻))
53	41, 49, 51, 52	leadd1dd 10520	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ ((#‘𝐻) + (𝑀 − 1)))
54	53, 10	syl6breqr 4625	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ 𝑁)
55	40, 54	eqbrtrd 4605	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ≤ 𝑁)
56	18	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ∈ ℤ)
57		hashcl 13009	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 ∈ Fin → (#‘𝐻) ∈ ℕ₀)
58		nn0z 11277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐻) ∈ ℕ₀ → (#‘𝐻) ∈ ℤ)
59	44, 57, 58	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘𝐻) ∈ ℤ)
60	59, 27	zaddcld 11362	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((#‘𝐻) + (𝑀 − 1)) ∈ ℤ)
61	10, 60	syl5eqel 2692	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ ℤ)
63		eluz 11577	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
64	56, 62, 63	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑁 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
65	55, 64	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ (ℤ_≥‘𝑀))
66		hashfz 13074	. . . . . . . . . . . . 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 6559	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 − 𝑀) = (((#‘𝐻) + (𝑀 − 1)) − 𝑀)
69	44, 57	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ₀)
70	69	nn0cnd 11230	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (#‘𝐻) ∈ ℂ)
71	70, 21, 19	addsubassd 10291	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((#‘𝐻) + (𝑀 − 1)) − 𝑀) = ((#‘𝐻) + ((𝑀 − 1) − 𝑀)))
72	68, 71	syl5eq 2656	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 𝑀) = ((#‘𝐻) + ((𝑀 − 1) − 𝑀)))
73	19, 20	negsubd 10277	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + -1) = (𝑀 − 1))
74	73	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) = (𝑀 + -1))
75	74	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = ((𝑀 + -1) − 𝑀))
76	20	negcld 10258	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → -1 ∈ ℂ)
77	19, 76	pncan2d 10273	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 + -1) − 𝑀) = -1)
78	75, 77	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = -1)
79	78	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐻) + ((𝑀 − 1) − 𝑀)) = ((#‘𝐻) + -1))
80	72, 79	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 𝑀) = ((#‘𝐻) + -1))
81	80	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) = (((#‘𝐻) + -1) + 1))
82	81	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((𝑁 − 𝑀) + 1) = (((#‘𝐻) + -1) + 1))
83	70, 20	negsubd 10277	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐻) + -1) = ((#‘𝐻) − 1))
84	83	oveq1d 6564	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((#‘𝐻) + -1) + 1) = (((#‘𝐻) − 1) + 1))
85	70, 20	npcand 10275	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((#‘𝐻) − 1) + 1) = (#‘𝐻))
86	84, 85	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((#‘𝐻) + -1) + 1) = (#‘𝐻))
87	86	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (((#‘𝐻) + -1) + 1) = (#‘𝐻))
88	67, 82, 87	3eqtrd 2648	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (#‘(𝑀...𝑁)) = (#‘𝐻))
89	37, 88	pm2.61dan 828	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝑀...𝑁)) = (#‘𝐻))
90	89	oveq2d 6565	. . . . . . . . 9 ⊢ (𝜑 → (1...(#‘(𝑀...𝑁))) = (1...(#‘𝐻)))
91		isoeq4 6470	. . . . . . . . 9 ⊢ ((1...(#‘(𝑀...𝑁))) = (1...(#‘𝐻)) → (ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁)) ↔ ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁))))
92	90, 91	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁)) ↔ ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁))))
93	92	biimpd 218	. . . . . . 7 ⊢ (𝜑 → (ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁)) → ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁))))
94	93	eximdv 1833	. . . . . 6 ⊢ (𝜑 → (∃ℎ ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁)) → ∃ℎ ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁))))
95	9, 94	mpi 20	. . . . 5 ⊢ (𝜑 → ∃ℎ ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)))
96		fzisoeu.or	. . . . . 6 ⊢ (𝜑 → < Or 𝐻)
97		fz1iso 13103	. . . . . 6 ⊢ (( < Or 𝐻 ∧ 𝐻 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))
98	96, 44, 97	syl2anc 691	. . . . 5 ⊢ (𝜑 → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))
99		eeanv 2170	. . . . 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)))
100	95, 98, 99	sylanbrc 695	. . . 4 ⊢ (𝜑 → ∃ℎ∃𝑔(ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻)))
101		isocnv 6480	. . . . . . . 8 ⊢ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(#‘𝐻))))
102	101	ad2antrl 760	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(#‘𝐻))))
103		simprr 792	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))) → 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))
104		isotr 6486	. . . . . . 7 ⊢ ((^◡ℎ Isom < , < ((𝑀...𝑁), (1...(#‘𝐻))) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻)) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
105	102, 103, 104	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
106	105	ex 449	. . . . 5 ⊢ (𝜑 → ((ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻)) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
107	106	2eximdv 1835	. . . 4 ⊢ (𝜑 → (∃ℎ∃𝑔(ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻)) → ∃ℎ∃𝑔(𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
108	100, 107	mpd 15	. . 3 ⊢ (𝜑 → ∃ℎ∃𝑔(𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
109		vex 3176	. . . . . . 7 ⊢ 𝑔 ∈ V
110		vex 3176	. . . . . . . 8 ⊢ ℎ ∈ V
111	110	cnvex 7006	. . . . . . 7 ⊢ ^◡ℎ ∈ V
112	109, 111	coex 7011	. . . . . 6 ⊢ (𝑔 ∘ ^◡ℎ) ∈ V
113		isoeq1 6467	. . . . . 6 ⊢ (𝑓 = (𝑔 ∘ ^◡ℎ) → (𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
114	112, 113	spcev 3273	. . . . 5 ⊢ ((𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
115	114	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)))
116	115	exlimdvv 1849	. . 3 ⊢ (𝜑 → (∃ℎ∃𝑔(𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)))
117	108, 116	mpd 15	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
118		ltwefz 12624	. . 3 ⊢ < We (𝑀...𝑁)
119		wemoiso 7044	. . 3 ⊢ ( < We (𝑀...𝑁) → ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
120	118, 119	mp1i 13	. 2 ⊢ (𝜑 → ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))
121		eu5 2484	. 2 ⊢ (∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ∧ ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)))
122	117, 120, 121	sylanbrc 695	1 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 ∃*wmo 2459 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 Or wor 4958 We wwe 4996 ^◡ccnv 5037 ∘ ccom 5042 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 Fincfn 7841 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980

This theorem is referenced by: fourierdlem36 39036

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