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Theorem isismt 25643
Description: Property of being an isometry. Compare with isismty 33911. (Contributed by Thierry Arnoux, 13-Dec-2019.)
Hypotheses
Ref Expression
isismt.b 𝐵 = (Base‘𝐺)
isismt.p 𝑃 = (Base‘𝐻)
isismt.d 𝐷 = (dist‘𝐺)
isismt.m = (dist‘𝐻)
Assertion
Ref Expression
isismt ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
Distinct variable groups:   𝐵,𝑎,𝑏   𝐹,𝑎,𝑏   𝐺,𝑎,𝑏   𝐻,𝑎,𝑏
Allowed substitution hints:   𝐷(𝑎,𝑏)   𝑃(𝑎,𝑏)   (𝑎,𝑏)   𝑉(𝑎,𝑏)   𝑊(𝑎,𝑏)

Proof of Theorem isismt
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3406 . . . 4 (𝐺𝑉𝐺 ∈ V)
2 elex 3406 . . . 4 (𝐻𝑊𝐻 ∈ V)
3 eqidd 2807 . . . . . . . 8 (𝑔 = 𝐺𝑓 = 𝑓)
4 fveq2 6408 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 isismt.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
64, 5syl6eqr 2858 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7 eqidd 2807 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘) = (Base‘))
83, 6, 7f1oeq123d 6349 . . . . . . 7 (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto→(Base‘)))
9 fveq2 6408 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
10 isismt.d . . . . . . . . . . . 12 𝐷 = (dist‘𝐺)
119, 10syl6eqr 2858 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
1211oveqd 6891 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
1312eqeq2d 2816 . . . . . . . . 9 (𝑔 = 𝐺 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
146, 13raleqbidv 3341 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
156, 14raleqbidv 3341 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
168, 15anbi12d 618 . . . . . 6 (𝑔 = 𝐺 → ((𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏)) ↔ (𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏))))
1716abbidv 2925 . . . . 5 (𝑔 = 𝐺 → {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏))} = {𝑓 ∣ (𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏))})
18 eqidd 2807 . . . . . . . 8 ( = 𝐻𝑓 = 𝑓)
19 eqidd 2807 . . . . . . . 8 ( = 𝐻𝐵 = 𝐵)
20 fveq2 6408 . . . . . . . . 9 ( = 𝐻 → (Base‘) = (Base‘𝐻))
21 isismt.p . . . . . . . . 9 𝑃 = (Base‘𝐻)
2220, 21syl6eqr 2858 . . . . . . . 8 ( = 𝐻 → (Base‘) = 𝑃)
2318, 19, 22f1oeq123d 6349 . . . . . . 7 ( = 𝐻 → (𝑓:𝐵1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto𝑃))
24 fveq2 6408 . . . . . . . . . . 11 ( = 𝐻 → (dist‘) = (dist‘𝐻))
25 isismt.m . . . . . . . . . . 11 = (dist‘𝐻)
2624, 25syl6eqr 2858 . . . . . . . . . 10 ( = 𝐻 → (dist‘) = )
2726oveqd 6891 . . . . . . . . 9 ( = 𝐻 → ((𝑓𝑎)(dist‘)(𝑓𝑏)) = ((𝑓𝑎) (𝑓𝑏)))
2827eqeq1d 2808 . . . . . . . 8 ( = 𝐻 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
29282ralbidv 3177 . . . . . . 7 ( = 𝐻 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
3023, 29anbi12d 618 . . . . . 6 ( = 𝐻 → ((𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))))
3130abbidv 2925 . . . . 5 ( = 𝐻 → {𝑓 ∣ (𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏))} = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
32 df-ismt 25642 . . . . 5 Ismt = (𝑔 ∈ V, ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏))})
33 ovex 6906 . . . . . 6 (𝑃𝑚 𝐵) ∈ V
34 f1of 6353 . . . . . . . . 9 (𝑓:𝐵1-1-onto𝑃𝑓:𝐵𝑃)
3521fvexi 6422 . . . . . . . . . 10 𝑃 ∈ V
365fvexi 6422 . . . . . . . . . 10 𝐵 ∈ V
3735, 36elmap 8121 . . . . . . . . 9 (𝑓 ∈ (𝑃𝑚 𝐵) ↔ 𝑓:𝐵𝑃)
3834, 37sylibr 225 . . . . . . . 8 (𝑓:𝐵1-1-onto𝑃𝑓 ∈ (𝑃𝑚 𝐵))
3938adantr 468 . . . . . . 7 ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃𝑚 𝐵))
4039abssi 3874 . . . . . 6 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃𝑚 𝐵)
4133, 40ssexi 4998 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ∈ V
4217, 31, 32, 41ovmpt2 7026 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
431, 2, 42syl2an 585 . . 3 ((𝐺𝑉𝐻𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
4443eleq2d 2871 . 2 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))}))
45 f1of 6353 . . . . 5 (𝐹:𝐵1-1-onto𝑃𝐹:𝐵𝑃)
46 fex 6714 . . . . 5 ((𝐹:𝐵𝑃𝐵 ∈ V) → 𝐹 ∈ V)
4745, 36, 46sylancl 576 . . . 4 (𝐹:𝐵1-1-onto𝑃𝐹 ∈ V)
4847adantr 468 . . 3 ((𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
49 f1oeq1 6343 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐵1-1-onto𝑃𝐹:𝐵1-1-onto𝑃))
50 fveq1 6407 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑎) = (𝐹𝑎))
51 fveq1 6407 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑏) = (𝐹𝑏))
5250, 51oveq12d 6892 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑎) (𝑓𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
5352eqeq1d 2808 . . . . 5 (𝑓 = 𝐹 → (((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
54532ralbidv 3177 . . . 4 (𝑓 = 𝐹 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5549, 54anbi12d 618 . . 3 (𝑓 = 𝐹 → ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
5648, 55elab3 3553 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5744, 56syl6bb 278 1 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  {cab 2792  wral 3096  Vcvv 3391  wf 6097  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6874  𝑚 cmap 8092  Basecbs 16068  distcds 16162  Ismtcismt 25641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-map 8094  df-ismt 25642
This theorem is referenced by:  ismot  25644
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