Theorem ismtycnv 37762

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 6874	. . . . 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 7321	. . . . . . . . . . 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 7321	. . . . . . . . . . 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 608	. . . . . . . . 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 7455	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝑥𝑀𝑦) = ((◡𝐹‘𝑢)𝑀𝑦))
11		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑢)))
12	11	oveq1d 7463	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)))
13	10, 12	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦))))
14		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((◡𝐹‘𝑢)𝑀𝑦) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
15		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝐹‘𝑣) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑣)))
16	15	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
17	14, 16	eqeq12d 2756	. . . . . . . . . 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 3646	. . . . . . . . 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 713	. . . . . . 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 7313	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
23	22	adantrr 716	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
24		f1ocnvfv2 7313	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
25	24	adantrl 715	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
26	23, 25	oveq12d 7466	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣))
27	26	adantlr 714	. . . . . 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 2781	. . . . 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 3208	. . . 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 37761	. 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 37761	. . 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 1537   ∈ wcel 2108  ∀wral 3067  ^◡ccnv 5699  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  ∞Metcxmet 21372   Ismty cismty 37758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-xr 11328  df-xmet 21380  df-ismty 37759

This theorem is referenced by:  ismtyhmeolem  37764  ismtyhmeo  37765  ismtybnd  37767