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Theorem ismtycnv 36261
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.
StepHypRef Expression
1 f1ocnv 6796 . . . . 5 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
21adantr 481 . . . 4 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → 𝐹:𝑌1-1-onto𝑋)
3 f1ocnvdm 7231 . . . . . . . . . . 11 ((𝐹:𝑋1-1-onto𝑌𝑢𝑌) → (𝐹𝑢) ∈ 𝑋)
43ex 413 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌 → (𝑢𝑌 → (𝐹𝑢) ∈ 𝑋))
5 f1ocnvdm 7231 . . . . . . . . . . 11 ((𝐹:𝑋1-1-onto𝑌𝑣𝑌) → (𝐹𝑣) ∈ 𝑋)
65ex 413 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌 → (𝑣𝑌 → (𝐹𝑣) ∈ 𝑋))
74, 6anim12d 609 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌 → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
87adantr 481 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
98imdistani 569 . . . . . . 7 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
10 oveq1 7364 . . . . . . . . . . 11 (𝑥 = (𝐹𝑢) → (𝑥𝑀𝑦) = ((𝐹𝑢)𝑀𝑦))
11 fveq2 6842 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑢) → (𝐹𝑥) = (𝐹‘(𝐹𝑢)))
1211oveq1d 7372 . . . . . . . . . . 11 (𝑥 = (𝐹𝑢) → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)))
1310, 12eqeq12d 2752 . . . . . . . . . 10 (𝑥 = (𝐹𝑢) → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ ((𝐹𝑢)𝑀𝑦) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦))))
14 oveq2 7365 . . . . . . . . . . 11 (𝑦 = (𝐹𝑣) → ((𝐹𝑢)𝑀𝑦) = ((𝐹𝑢)𝑀(𝐹𝑣)))
15 fveq2 6842 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑣) → (𝐹𝑦) = (𝐹‘(𝐹𝑣)))
1615oveq2d 7373 . . . . . . . . . . 11 (𝑦 = (𝐹𝑣) → ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
1714, 16eqeq12d 2752 . . . . . . . . . 10 (𝑦 = (𝐹𝑣) → (((𝐹𝑢)𝑀𝑦) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)) ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣)))))
1813, 17rspc2v 3590 . . . . . . . . 9 (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣)))))
1918impcom 408 . . . . . . . 8 ((∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
2019adantll 712 . . . . . . 7 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
219, 20syl 17 . . . . . 6 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
22 f1ocnvfv2 7223 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑢𝑌) → (𝐹‘(𝐹𝑢)) = 𝑢)
2322adantrr 715 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝐹‘(𝐹𝑢)) = 𝑢)
24 f1ocnvfv2 7223 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑣𝑌) → (𝐹‘(𝐹𝑣)) = 𝑣)
2524adantrl 714 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝐹‘(𝐹𝑣)) = 𝑣)
2623, 25oveq12d 7375 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))) = (𝑢𝑁𝑣))
2726adantlr 713 . . . . . 6 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))) = (𝑢𝑁𝑣))
2821, 27eqtr2d 2777 . . . . 5 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
2928ralrimivva 3197 . . . 4 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
302, 29jca 512 . . 3 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣))))
3130a1i 11 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))))
32 isismty 36260 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
33 isismty 36260 . . 3 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑀 ∈ (∞Met‘𝑋)) → (𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))))
3433ancoms 459 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))))
3531, 32, 343imtr4d 293 1 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → 𝐹 ∈ (𝑁 Ismty 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  ccnv 5632  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  ∞Metcxmet 20781   Ismty cismty 36257
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-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  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-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-xr 11193  df-xmet 20789  df-ismty 36258
This theorem is referenced by:  ismtyhmeolem  36263  ismtyhmeo  36264  ismtybnd  36266
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