fzisoeu - Mathbox for Glauco Siliprandi

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Theorem fzisoeu 45250

Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 14497 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 13562	. . . . . . . . 9 ⊢ (𝑀...𝑁) ⊆ ℤ
2		zssre 12617	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
3	1, 2	sstri 4004	. . . . . . . 8 ⊢ (𝑀...𝑁) ⊆ ℝ
4		ltso 11338	. . . . . . . 8 ⊢ < Or ℝ
5		soss 5616	. . . . . . . 8 ⊢ ((𝑀...𝑁) ⊆ ℝ → ( < Or ℝ → < Or (𝑀...𝑁)))
6	3, 4, 5	mp2 9	. . . . . . 7 ⊢ < Or (𝑀...𝑁)
7		fzfi 14009	. . . . . . 7 ⊢ (𝑀...𝑁) ∈ Fin
8		fz1iso 14497	. . . . . . 7 ⊢ (( < Or (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → ∃ℎ ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)))
9	6, 7, 8	mp2an 692	. . . . . 6 ⊢ ∃ℎ ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁))
10		fzisoeu.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 = ((♯‘𝐻) + (𝑀 − 1))
11		fveq2 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐻 = ∅ → (♯‘𝐻) = (♯‘∅))
12		hash0 14402	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘∅) = 0
13	11, 12	eqtrdi 2790	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 = ∅ → (♯‘𝐻) = 0)
14	13	oveq1d 7445	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 = ∅ → ((♯‘𝐻) + (𝑀 − 1)) = (0 + (𝑀 − 1)))
15	10, 14	eqtrid 2786	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ → 𝑁 = (0 + (𝑀 − 1)))
16	15	oveq2d 7446	. . . . . . . . . . . . . 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 12720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℂ)
20		1cnd 11253	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℂ)
21	19, 20	subcld 11617	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℂ)
22	21	addlidd 11459	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + (𝑀 − 1)) = (𝑀 − 1))
23	22	oveq2d 7446	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = (𝑀...(𝑀 − 1)))
24	18	zred 12719	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	24	ltm1d 12197	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
26		peano2zm 12657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
27	18, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
28		fzn 13576	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
31	23, 30	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = ∅)
32	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...(0 + (𝑀 − 1))) = ∅)
33		eqcom 2741	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ ↔ ∅ = 𝐻)
34	33	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝐻 = ∅ → ∅ = 𝐻)
35	34	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → ∅ = 𝐻)
36	17, 32, 35	3eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = 𝐻)
37	36	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
38	20, 19	pncan3d 11620	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 + (𝑀 − 1)) = 𝑀)
39	38	eqcomd 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 = (1 + (𝑀 − 1)))
40	39	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 = (1 + (𝑀 − 1)))
41		1red 11259	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ∈ ℝ)
42		neqne 2945	. . . . . . . . . . . . . . . . . . . 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 14401	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 ∈ Fin → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
47	45, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
48	43, 47	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘𝐻) ∈ ℕ)
49	48	nnred 12278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘𝐻) ∈ ℝ)
50	27	zred 12719	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
51	50	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑀 − 1) ∈ ℝ)
52	48	nnge1d 12311	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ≤ (♯‘𝐻))
53	41, 49, 51, 52	leadd1dd 11874	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ ((♯‘𝐻) + (𝑀 − 1)))
54	53, 10	breqtrrdi 5189	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ 𝑁)
55	40, 54	eqbrtrd 5169	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ≤ 𝑁)
56	18	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ∈ ℤ)
57		hashcl 14391	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 ∈ Fin → (♯‘𝐻) ∈ ℕ₀)
58		nn0z 12635	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐻) ∈ ℕ₀ → (♯‘𝐻) ∈ ℤ)
59	44, 57, 58	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝐻) ∈ ℤ)
60	59, 27	zaddcld 12723	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝐻) + (𝑀 − 1)) ∈ ℤ)
61	10, 60	eqeltrid 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ ℤ)
63		eluz 12889	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ (ℤ_≥‘𝑀))
66		hashfz 14462	. . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 − 𝑀) = (((♯‘𝐻) + (𝑀 − 1)) − 𝑀)
69	44, 57	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝐻) ∈ ℕ₀)
70	69	nn0cnd 12586	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝐻) ∈ ℂ)
71	70, 21, 19	addsubassd 11637	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((♯‘𝐻) + (𝑀 − 1)) − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)))
72	68, 71	eqtrid 2786	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)))
73	20	negcld 11604	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → -1 ∈ ℂ)
74	19, 20	negsubd 11623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + -1) = (𝑀 − 1))
75	74	eqcomd 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) = (𝑀 + -1))
76	19, 73, 75	mvrladdd 11673	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = -1)
77	76	oveq2d 7446	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)) = ((♯‘𝐻) + -1))
78	72, 77	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + -1))
79	78	oveq1d 7445	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) = (((♯‘𝐻) + -1) + 1))
80	79	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((𝑁 − 𝑀) + 1) = (((♯‘𝐻) + -1) + 1))
81	70, 20	negsubd 11623	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐻) + -1) = ((♯‘𝐻) − 1))
82	81	oveq1d 7445	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) = (((♯‘𝐻) − 1) + 1))
83	70, 20	npcand 11621	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝐻) − 1) + 1) = (♯‘𝐻))
84	82, 83	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) = (♯‘𝐻))
85	84	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (((♯‘𝐻) + -1) + 1) = (♯‘𝐻))
86	67, 80, 85	3eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
87	37, 86	pm2.61dan 813	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
88	87	oveq2d 7446	. . . . . . . . 9 ⊢ (𝜑 → (1...(♯‘(𝑀...𝑁))) = (1...(♯‘𝐻)))
89		isoeq4 7339	. . . . . . . . 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 229	. . . . . . 7 ⊢ (𝜑 → (ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) → ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁))))
92	91	eximdv 1914	. . . . . 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 14497	. . . . . 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 1952	. . . . 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 7349	. . . . . . . 8 ⊢ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻))))
100	99	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻))))
101		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))
102		isotr 7355	. . . . . . 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 412	. . . . 5 ⊢ (𝜑 → ((ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
105	104	2eximdv 1916	. . . 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 3481	. . . . . . 7 ⊢ 𝑔 ∈ V
108		vex 3481	. . . . . . . 8 ⊢ ℎ ∈ V
109	108	cnvex 7947	. . . . . . 7 ⊢ ^◡ℎ ∈ V
110	107, 109	coex 7952	. . . . . 6 ⊢ (𝑔 ∘ ^◡ℎ) ∈ V
111		isoeq1 7336	. . . . . 6 ⊢ (𝑓 = (𝑔 ∘ ^◡ℎ) → (𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
112	110, 111	spcev 3605	. . . . 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 1931	. . 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 14000	. . 3 ⊢ < We (𝑀...𝑁)
117		wemoiso 7996	. . 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 2566	. 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 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∃*wmo 2535 ∃!weu 2565 ≠ wne 2937 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 Or wor 5595 We wwe 5639 ^◡ccnv 5687 ∘ ccom 5692 ‘cfv 6562 Isom wiso 6563 (class class class)co 7430 Fincfn 8983 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 < clt 11292 ≤ cle 11293 − cmin 11489 -cneg 11490 ℕcn 12263 ℕ₀cn0 12523 ℤcz 12610 ℤ_≥cuz 12875 ...cfz 13543 ♯chash 14365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366

This theorem is referenced by: fourierdlem36 46098

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