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Theorem ismtycnv 37789
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 6861 . . . . 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 7305 . . . . . . . . . . 11 ((𝐹:𝑋1-1-onto𝑌𝑢𝑌) → (𝐹𝑢) ∈ 𝑋)
43ex 412 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌 → (𝑢𝑌 → (𝐹𝑢) ∈ 𝑋))
5 f1ocnvdm 7305 . . . . . . . . . . 11 ((𝐹:𝑋1-1-onto𝑌𝑣𝑌) → (𝐹𝑣) ∈ 𝑋)
65ex 412 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌 → (𝑣𝑌 → (𝐹𝑣) ∈ 𝑋))
74, 6anim12d 609 . . . . . . . . 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 568 . . . . . . 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 7438 . . . . . . . . . . 11 (𝑥 = (𝐹𝑢) → (𝑥𝑀𝑦) = ((𝐹𝑢)𝑀𝑦))
11 fveq2 6907 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑢) → (𝐹𝑥) = (𝐹‘(𝐹𝑢)))
1211oveq1d 7446 . . . . . . . . . . 11 (𝑥 = (𝐹𝑢) → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)))
1310, 12eqeq12d 2751 . . . . . . . . . 10 (𝑥 = (𝐹𝑢) → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ ((𝐹𝑢)𝑀𝑦) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦))))
14 oveq2 7439 . . . . . . . . . . 11 (𝑦 = (𝐹𝑣) → ((𝐹𝑢)𝑀𝑦) = ((𝐹𝑢)𝑀(𝐹𝑣)))
15 fveq2 6907 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑣) → (𝐹𝑦) = (𝐹‘(𝐹𝑣)))
1615oveq2d 7447 . . . . . . . . . . 11 (𝑦 = (𝐹𝑣) → ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
1714, 16eqeq12d 2751 . . . . . . . . . 10 (𝑦 = (𝐹𝑣) → (((𝐹𝑢)𝑀𝑦) = ((𝐹‘(𝐹𝑢))𝑁(𝐹𝑦)) ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣)))))
1813, 17rspc2v 3633 . . . . . . . . 9 (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣)))))
1918impcom 407 . . . . . . . 8 ((∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))))
2019adantll 714 . . . . . . 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 7297 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑢𝑌) → (𝐹‘(𝐹𝑢)) = 𝑢)
2322adantrr 717 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝐹‘(𝐹𝑢)) = 𝑢)
24 f1ocnvfv2 7297 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑣𝑌) → (𝐹‘(𝐹𝑣)) = 𝑣)
2524adantrl 716 . . . . . . . 8 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝐹‘(𝐹𝑣)) = 𝑣)
2623, 25oveq12d 7449 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))) = (𝑢𝑁𝑣))
2726adantlr 715 . . . . . 6 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹‘(𝐹𝑢))𝑁(𝐹‘(𝐹𝑣))) = (𝑢𝑁𝑣))
2821, 27eqtr2d 2776 . . . . 5 (((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
2928ralrimivva 3200 . . . 4 ((𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
302, 29jca 511 . . 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 37788 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
33 isismty 37788 . . 3 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑀 ∈ (∞Met‘𝑋)) → (𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))))
3433ancoms 458 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (𝐹:𝑌1-1-onto𝑋 ∧ ∀𝑢𝑌𝑣𝑌 (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))))
3531, 32, 343imtr4d 294 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  ccnv 5688  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  ∞Metcxmet 21367   Ismty cismty 37785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-xr 11297  df-xmet 21375  df-ismty 37786
This theorem is referenced by:  ismtyhmeolem  37791  ismtyhmeo  37792  ismtybnd  37794
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