Theorem ismtycnv 37841

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 6775	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
2	1	adantr 480	. . . 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 7219	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (◡𝐹‘𝑢) ∈ 𝑋)
4	3	ex 412	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑢 ∈ 𝑌 → (◡𝐹‘𝑢) ∈ 𝑋))
5		f1ocnvdm 7219	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (◡𝐹‘𝑣) ∈ 𝑋)
6	5	ex 412	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑣 ∈ 𝑌 → (◡𝐹‘𝑣) ∈ 𝑋))
7	4, 6	anim12d 609	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)))
8	7	adantr 480	. . . . . . . 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 568	. . . . . . 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 7353	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝑥𝑀𝑦) = ((◡𝐹‘𝑢)𝑀𝑦))
11		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑢)))
12	11	oveq1d 7361	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)))
13	10, 12	eqeq12d 2747	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦))))
14		oveq2 7354	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((◡𝐹‘𝑢)𝑀𝑦) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
15		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝐹‘𝑣) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑣)))
16	15	oveq2d 7362	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
17	14, 16	eqeq12d 2747	. . . . . . . . . 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 3583	. . . . . . . . 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 407	. . . . . . . 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 714	. . . . . . 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 17	. . . . . 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 7211	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
23	22	adantrr 717	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
24		f1ocnvfv2 7211	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
25	24	adantrl 716	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
26	23, 25	oveq12d 7364	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣))
27	26	adantlr 715	. . . . . 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 2767	. . . . 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 3175	. . . 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 511	. . 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 37840	. 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 37840	. . 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 458	. 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 294	1 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ^◡ccnv 5613  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  ∞Metcxmet 21276   Ismty cismty 37837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-xr 11150  df-xmet 21284  df-ismty 37838

This theorem is referenced by:  ismtyhmeolem  37843  ismtyhmeo  37844  ismtybnd  37846