fzisoeu - Mathbox for Glauco Siliprandi

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Theorem fzisoeu 43524

Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 14361 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 13443	. . . . . . . . 9 ⊢ (𝑀...𝑁) ⊆ ℤ
2		zssre 12506	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
3	1, 2	sstri 3953	. . . . . . . 8 ⊢ (𝑀...𝑁) ⊆ ℝ
4		ltso 11235	. . . . . . . 8 ⊢ < Or ℝ
5		soss 5565	. . . . . . . 8 ⊢ ((𝑀...𝑁) ⊆ ℝ → ( < Or ℝ → < Or (𝑀...𝑁)))
6	3, 4, 5	mp2 9	. . . . . . 7 ⊢ < Or (𝑀...𝑁)
7		fzfi 13877	. . . . . . 7 ⊢ (𝑀...𝑁) ∈ Fin
8		fz1iso 14361	. . . . . . 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 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐻 = ∅ → (♯‘𝐻) = (♯‘∅))
12		hash0 14267	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘∅) = 0
13	11, 12	eqtrdi 2792	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 = ∅ → (♯‘𝐻) = 0)
14	13	oveq1d 7372	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 = ∅ → ((♯‘𝐻) + (𝑀 − 1)) = (0 + (𝑀 − 1)))
15	10, 14	eqtrid 2788	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ → 𝑁 = (0 + (𝑀 − 1)))
16	15	oveq2d 7373	. . . . . . . . . . . . . 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 12608	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℂ)
20		1cnd 11150	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℂ)
21	19, 20	subcld 11512	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℂ)
22	21	addid2d 11356	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + (𝑀 − 1)) = (𝑀 − 1))
23	22	oveq2d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = (𝑀...(𝑀 − 1)))
24	18	zred 12607	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	24	ltm1d 12087	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
26		peano2zm 12546	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
27	18, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
28		fzn 13457	. . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = ∅)
32	31	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...(0 + (𝑀 − 1))) = ∅)
33		eqcom 2743	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ ↔ ∅ = 𝐻)
34	33	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (𝐻 = ∅ → ∅ = 𝐻)
35	34	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → ∅ = 𝐻)
36	17, 32, 35	3eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = 𝐻)
37	36	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
38	20, 19	pncan3d 11515	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 + (𝑀 − 1)) = 𝑀)
39	38	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 = (1 + (𝑀 − 1)))
40	39	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 = (1 + (𝑀 − 1)))
41		1red 11156	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ∈ ℝ)
42		neqne 2951	. . . . . . . . . . . . . . . . . . . 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 14266	. . . . . . . . . . . . . . . . . . . 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 12168	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘𝐻) ∈ ℝ)
50	27	zred 12607	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
51	50	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑀 − 1) ∈ ℝ)
52	48	nnge1d 12201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ≤ (♯‘𝐻))
53	41, 49, 51, 52	leadd1dd 11769	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ ((♯‘𝐻) + (𝑀 − 1)))
54	53, 10	breqtrrdi 5147	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ 𝑁)
55	40, 54	eqbrtrd 5127	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ≤ 𝑁)
56	18	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ∈ ℤ)
57		hashcl 14256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 ∈ Fin → (♯‘𝐻) ∈ ℕ₀)
58		nn0z 12524	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐻) ∈ ℕ₀ → (♯‘𝐻) ∈ ℤ)
59	44, 57, 58	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝐻) ∈ ℤ)
60	59, 27	zaddcld 12611	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝐻) + (𝑀 − 1)) ∈ ℤ)
61	10, 60	eqeltrid 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ ℤ)
63		eluz 12777	. . . . . . . . . . . . . . 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 14327	. . . . . . . . . . . . 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 7367	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 − 𝑀) = (((♯‘𝐻) + (𝑀 − 1)) − 𝑀)
69	44, 57	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝐻) ∈ ℕ₀)
70	69	nn0cnd 12475	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝐻) ∈ ℂ)
71	70, 21, 19	addsubassd 11532	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((♯‘𝐻) + (𝑀 − 1)) − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)))
72	68, 71	eqtrid 2788	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)))
73	20	negcld 11499	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → -1 ∈ ℂ)
74	19, 20	negsubd 11518	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + -1) = (𝑀 − 1))
75	74	eqcomd 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) = (𝑀 + -1))
76	19, 73, 75	mvrladdd 11568	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = -1)
77	76	oveq2d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)) = ((♯‘𝐻) + -1))
78	72, 77	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + -1))
79	78	oveq1d 7372	. . . . . . . . . . . . 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 11518	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐻) + -1) = ((♯‘𝐻) − 1))
82	81	oveq1d 7372	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) = (((♯‘𝐻) − 1) + 1))
83	70, 20	npcand 11516	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝐻) − 1) + 1) = (♯‘𝐻))
84	82, 83	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) = (♯‘𝐻))
85	84	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (((♯‘𝐻) + -1) + 1) = (♯‘𝐻))
86	67, 80, 85	3eqtrd 2780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
87	37, 86	pm2.61dan 811	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑀...𝑁)) = (♯‘𝐻))
88	87	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → (1...(♯‘(𝑀...𝑁))) = (1...(♯‘𝐻)))
89		isoeq4 7265	. . . . . . . . 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 14361	. . . . . 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 7275	. . . . . . . 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 7281	. . . . . . 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 3449	. . . . . . 7 ⊢ 𝑔 ∈ V
108		vex 3449	. . . . . . . 8 ⊢ ℎ ∈ V
109	108	cnvex 7862	. . . . . . 7 ⊢ ^◡ℎ ∈ V
110	107, 109	coex 7867	. . . . . 6 ⊢ (𝑔 ∘ ^◡ℎ) ∈ V
111		isoeq1 7262	. . . . . 6 ⊢ (𝑓 = (𝑔 ∘ ^◡ℎ) → (𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
112	110, 111	spcev 3565	. . . . 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 13868	. . 3 ⊢ < We (𝑀...𝑁)
117		wemoiso 7906	. . 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 2567	. 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 2536 ∃!weu 2566 ≠ wne 2943 ⊆ wss 3910 ∅c0 4282 class class class wbr 5105 Or wor 5544 We wwe 5587 ^◡ccnv 5632 ∘ ccom 5637 ‘cfv 6496 Isom wiso 6497 (class class class)co 7357 Fincfn 8883 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 -cneg 11386 ℕcn 12153 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ...cfz 13424 ♯chash 14230

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231

This theorem is referenced by: fourierdlem36 44374

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