fzisoeu - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fzisoeu		Structured version Visualization version GIF version

Theorem fzisoeu 39514

Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13246 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 12343	. . . . . . . . 9 ⊢ (𝑀...𝑁) ⊆ ℤ
2		zssre 11384	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
3	1, 2	sstri 3612	. . . . . . . 8 ⊢ (𝑀...𝑁) ⊆ ℝ
4		ltso 10118	. . . . . . . 8 ⊢ < Or ℝ
5		soss 5053	. . . . . . . 8 ⊢ ((𝑀...𝑁) ⊆ ℝ → ( < Or ℝ → < Or (𝑀...𝑁)))
6	3, 4, 5	mp2 9	. . . . . . 7 ⊢ < Or (𝑀...𝑁)
7		fzfi 12771	. . . . . . 7 ⊢ (𝑀...𝑁) ∈ Fin
8		fz1iso 13246	. . . . . . 7 ⊢ (( < Or (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → ∃ℎ ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁)))
9	6, 7, 8	mp2an 708	. . . . . 6 ⊢ ∃ℎ ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁))
10		fzisoeu.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 = ((#‘𝐻) + (𝑀 − 1))
11		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐻 = ∅ → (#‘𝐻) = (#‘∅))
12		hash0 13158	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘∅) = 0
13	11, 12	syl6eq 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 = ∅ → (#‘𝐻) = 0)
14	13	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 = ∅ → ((#‘𝐻) + (𝑀 − 1)) = (0 + (𝑀 − 1)))
15	10, 14	syl5eq 2668	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ → 𝑁 = (0 + (𝑀 − 1)))
16	15	oveq2d 6666	. . . . . . . . . . . . . 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 11483	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℂ)
20		1cnd 10056	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℂ)
21	19, 20	subcld 10392	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℂ)
22	21	addid2d 10237	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + (𝑀 − 1)) = (𝑀 − 1))
23	22	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = (𝑀...(𝑀 − 1)))
24	18	zred 11482	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	24	ltm1d 10956	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
26		peano2zm 11420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
27	18, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
28		fzn 12357	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
29	18, 27, 28	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
30	25, 29	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
31	23, 30	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = ∅)
32	31	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...(0 + (𝑀 − 1))) = ∅)
33		eqcom 2629	. . . . . . . . . . . . . . 15 ⊢ (𝐻 = ∅ ↔ ∅ = 𝐻)
34	33	biimpi 206	. . . . . . . . . . . . . 14 ⊢ (𝐻 = ∅ → ∅ = 𝐻)
35	34	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 = ∅) → ∅ = 𝐻)
36	17, 32, 35	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = 𝐻)
37	36	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 = ∅) → (#‘(𝑀...𝑁)) = (#‘𝐻))
38	20, 19	pncan3d 10395	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 + (𝑀 − 1)) = 𝑀)
39	38	eqcomd 2628	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 = (1 + (𝑀 − 1)))
40	39	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 = (1 + (𝑀 − 1)))
41		1red 10055	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ∈ ℝ)
42		neqne 2802	. . . . . . . . . . . . . . . . . . . 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 13157	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 ∈ Fin → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
47	45, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
48	43, 47	mpbird 247	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (#‘𝐻) ∈ ℕ)
49	48	nnred 11035	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (#‘𝐻) ∈ ℝ)
50	27	zred 11482	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
51	50	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑀 − 1) ∈ ℝ)
52	48	nnge1d 11063	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ≤ (#‘𝐻))
53	41, 49, 51, 52	leadd1dd 10641	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ ((#‘𝐻) + (𝑀 − 1)))
54	53, 10	syl6breqr 4695	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ 𝑁)
55	40, 54	eqbrtrd 4675	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ≤ 𝑁)
56	18	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ∈ ℤ)
57		hashcl 13147	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 ∈ Fin → (#‘𝐻) ∈ ℕ₀)
58		nn0z 11400	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐻) ∈ ℕ₀ → (#‘𝐻) ∈ ℤ)
59	44, 57, 58	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘𝐻) ∈ ℤ)
60	59, 27	zaddcld 11486	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((#‘𝐻) + (𝑀 − 1)) ∈ ℤ)
61	10, 60	syl5eqel 2705	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ ℤ)
63		eluz 11701	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
64	56, 62, 63	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑁 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
65	55, 64	mpbird 247	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ (ℤ_≥‘𝑀))
66		hashfz 13214	. . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 − 𝑀) = (((#‘𝐻) + (𝑀 − 1)) − 𝑀)
69	44, 57	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ₀)
70	69	nn0cnd 11353	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (#‘𝐻) ∈ ℂ)
71	70, 21, 19	addsubassd 10412	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((#‘𝐻) + (𝑀 − 1)) − 𝑀) = ((#‘𝐻) + ((𝑀 − 1) − 𝑀)))
72	68, 71	syl5eq 2668	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 𝑀) = ((#‘𝐻) + ((𝑀 − 1) − 𝑀)))
73	19, 20	negsubd 10398	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + -1) = (𝑀 − 1))
74	73	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) = (𝑀 + -1))
75	74	oveq1d 6665	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = ((𝑀 + -1) − 𝑀))
76	20	negcld 10379	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → -1 ∈ ℂ)
77	19, 76	pncan2d 10394	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 + -1) − 𝑀) = -1)
78	75, 77	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = -1)
79	78	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐻) + ((𝑀 − 1) − 𝑀)) = ((#‘𝐻) + -1))
80	72, 79	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 𝑀) = ((#‘𝐻) + -1))
81	80	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) = (((#‘𝐻) + -1) + 1))
82	81	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((𝑁 − 𝑀) + 1) = (((#‘𝐻) + -1) + 1))
83	70, 20	negsubd 10398	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐻) + -1) = ((#‘𝐻) − 1))
84	83	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((#‘𝐻) + -1) + 1) = (((#‘𝐻) − 1) + 1))
85	70, 20	npcand 10396	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((#‘𝐻) − 1) + 1) = (#‘𝐻))
86	84, 85	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((#‘𝐻) + -1) + 1) = (#‘𝐻))
87	86	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (((#‘𝐻) + -1) + 1) = (#‘𝐻))
88	67, 82, 87	3eqtrd 2660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (#‘(𝑀...𝑁)) = (#‘𝐻))
89	37, 88	pm2.61dan 832	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝑀...𝑁)) = (#‘𝐻))
90	89	oveq2d 6666	. . . . . . . . 9 ⊢ (𝜑 → (1...(#‘(𝑀...𝑁))) = (1...(#‘𝐻)))
91		isoeq4 6570	. . . . . . . . 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 219	. . . . . . 7 ⊢ (𝜑 → (ℎ Isom < , < ((1...(#‘(𝑀...𝑁))), (𝑀...𝑁)) → ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁))))
94	93	eximdv 1846	. . . . . 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 13246	. . . . . 6 ⊢ (( < Or 𝐻 ∧ 𝐻 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))
98	96, 44, 97	syl2anc 693	. . . . 5 ⊢ (𝜑 → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))
99		eeanv 2182	. . . . 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 698	. . . 4 ⊢ (𝜑 → ∃ℎ∃𝑔(ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻)))
101		isocnv 6580	. . . . . . . 8 ⊢ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(#‘𝐻))))
102	101	ad2antrl 764	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))) → ^◡ℎ Isom < , < ((𝑀...𝑁), (1...(#‘𝐻))))
103		simprr 796	. . . . . . 7 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))) → 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))
104		isotr 6586	. . . . . . 7 ⊢ ((^◡ℎ Isom < , < ((𝑀...𝑁), (1...(#‘𝐻))) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻)) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
105	102, 103, 104	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻))) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))
106	105	ex 450	. . . . 5 ⊢ (𝜑 → ((ℎ Isom < , < ((1...(#‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(#‘𝐻)), 𝐻)) → (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
107	106	2eximdv 1848	. . . 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 3203	. . . . . . 7 ⊢ 𝑔 ∈ V
110		vex 3203	. . . . . . . 8 ⊢ ℎ ∈ V
111	110	cnvex 7113	. . . . . . 7 ⊢ ^◡ℎ ∈ V
112	109, 111	coex 7118	. . . . . 6 ⊢ (𝑔 ∘ ^◡ℎ) ∈ V
113		isoeq1 6567	. . . . . 6 ⊢ (𝑓 = (𝑔 ∘ ^◡ℎ) → (𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (𝑔 ∘ ^◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)))
114	112, 113	spcev 3300	. . . . 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 1862	. . 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 12762	. . 3 ⊢ < We (𝑀...𝑁)
119		wemoiso 7153	. . 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 2496	. 2 ⊢ (∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ∧ ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)))
122	117, 120, 121	sylanbrc 698	1 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 ∃*wmo 2471 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 Or wor 5034 We wwe 5072 ^◡ccnv 5113 ∘ ccom 5118 ‘cfv 5888 Isom wiso 5889 (class class class)co 6650 Fincfn 7955 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 #chash 13117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118

This theorem is referenced by: fourierdlem36 40360

W3C validator