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Theorem ismtycnv 33731
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 6187 . . . . 5 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
21adantr 480 . . . 4 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → 𝐹:𝑌1-1-onto𝑋)
3 f1ocnvdm 6580 . . . . . . . . . . 11 ((𝐹:𝑋1-1-onto𝑌𝑢𝑌) → (𝐹𝑢) ∈ 𝑋)
43ex 449 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌 → (𝑢𝑌 → (𝐹𝑢) ∈ 𝑋))
5 f1ocnvdm 6580 . . . . . . . . . . 11 ((𝐹:𝑋1-1-onto𝑌𝑣𝑌) → (𝐹𝑣) ∈ 𝑋)
65ex 449 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌 → (𝑣𝑌 → (𝐹𝑣) ∈ 𝑋))
74, 6anim12d 585 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌 → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
87adantr 480 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
98imdistani 726 . . . . . . 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 6697 . . . . . . . . . . 11 (𝑥 = (𝐹𝑢) → (𝑥𝑀𝑦) = ((𝐹𝑢)𝑀𝑦))
11 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑢) → (𝐹𝑥) = (𝐹‘(𝐹𝑢)))
1211oveq1d 6705 . . . . . . . . . . 11 (𝑥 = (𝐹𝑢) → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)))
1310, 12eqeq12d 2666 . . . . . . . . . 10 (𝑥 = (𝐹𝑢) → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ ((𝐹𝑢)𝑀𝑦) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦))))
14 oveq2 6698 . . . . . . . . . . 11 (𝑦 = (𝐹𝑣) → ((𝐹𝑢)𝑀𝑦) = ((𝐹𝑢)𝑀(𝐹𝑣)))
15 fveq2 6229 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑣) → (𝐹𝑦) = (𝐹‘(𝐹𝑣)))
1615oveq2d 6706 . . . . . . . . . . 11 (𝑦 = (𝐹𝑣) → ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
1714, 16eqeq12d 2666 . . . . . . . . . 10 (𝑦 = (𝐹𝑣) → (((𝐹𝑢)𝑀𝑦) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)) ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣)))))
1813, 17rspc2v 3353 . . . . . . . . 9 (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣)))))
1918impcom 445 . . . . . . . 8 ((∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
2019adantll 750 . . . . . . 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 6573 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑢𝑌) → (𝐹‘(𝐹𝑢)) = 𝑢)
2322adantrr 753 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝐹‘(𝐹𝑢)) = 𝑢)
24 f1ocnvfv2 6573 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑣𝑌) → (𝐹‘(𝐹𝑣)) = 𝑣)
2524adantrl 752 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝐹‘(𝐹𝑣)) = 𝑣)
2623, 25oveq12d 6708 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))) = (𝑢𝑁𝑣))
2726adantlr 751 . . . . . 6 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))) = (𝑢𝑁𝑣))
2821, 27eqtr2d 2686 . . . . 5 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
2928ralrimivva 3000 . . . 4 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
302, 29jca 553 . . 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 33730 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
33 isismty 33730 . . 3 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑀 ∈ (∞Met‘𝑋)) → (𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))))
3433ancoms 468 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))))
3531, 32, 343imtr4d 283 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  ccnv 5142  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  ∞Metcxmt 19779   Ismty cismty 33727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-xr 10116  df-xmet 19787  df-ismty 33728
This theorem is referenced by:  ismtyhmeolem  33733  ismtyhmeo  33734  ismtybnd  33736
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