Theorem ismtycnv 38481

Description: The inverse of an isometry is an isometry. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
ismtycnv	⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀)))

Proof of Theorem ismtycnv

Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ocnv 6833	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
2	1	adantr 485	. . . 4 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ◡𝐹:𝑌–1-1-onto→𝑋)
3		f1ocnvdm 7283	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (◡𝐹‘𝑢) ∈ 𝑋)
4	3	ex 417	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑢 ∈ 𝑌 → (◡𝐹‘𝑢) ∈ 𝑋))
5		f1ocnvdm 7283	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (◡𝐹‘𝑣) ∈ 𝑋)
6	5	ex 417	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑣 ∈ 𝑌 → (◡𝐹‘𝑣) ∈ 𝑋))
7	4, 6	anim12d 620	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)))
8	7	adantr 485	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)))
9	8	imdistani 578	. . . . . . 7 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)))
10		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝑥𝑀𝑦) = ((◡𝐹‘𝑢)𝑀𝑦))
11		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑢)))
12	11	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)))
13	10, 12	eqeq12d 2778	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦))))
14		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((◡𝐹‘𝑢)𝑀𝑦) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
15		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝐹‘𝑣) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑣)))
16	15	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
17	14, 16	eqeq12d 2778	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐹‘𝑣) → (((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))))
18	13, 17	rspc2v 3591	. . . . . . . . 9 ⊢ (((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))))
19	18	impcom 412	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
20	19	adantll 726	. . . . . . 7 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
21	9, 20	syl 18	. . . . . 6 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
22		f1ocnvfv2 7275	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
23	22	adantrr 729	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
24		f1ocnvfv2 7275	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
25	24	adantrl 728	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
26	23, 25	oveq12d 7430	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣))
27	26	adantlr 727	. . . . . 6 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣))
28	21, 27	eqtr2d 2798	. . . . 5 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
29	28	ralrimivva 3207	. . . 4 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
30	2, 29	jca 520	. . 3 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))))
31	30	a1i 11	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))))
32		isismty 38480	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))))
33		isismty 38480	. . 3 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑀 ∈ (∞Met‘𝑋)) → (◡𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))))
34	33	ancoms 463	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (◡𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))))
35	31, 32, 34	3imtr4d 297	1 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ^◡ccnv 5659  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7412  ∞Metcxmet 21518   Ismty cismty 38477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-xr 11253  df-xmet 21526  df-ismty 38478

This theorem is used by:  ismtyhmeolem  38483  ismtyhmeo  38484  ismtybnd  38486