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Theorem isismt 25336
Description: Property of being an isometry. Compare with isismty 33253. (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 3198 . . . 4 (𝐺𝑉𝐺 ∈ V)
2 elex 3198 . . . 4 (𝐻𝑊𝐻 ∈ V)
3 eqidd 2622 . . . . . . . 8 (𝑔 = 𝐺𝑓 = 𝑓)
4 fveq2 6150 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 isismt.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
64, 5syl6eqr 2673 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7 eqidd 2622 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘) = (Base‘))
83, 6, 7f1oeq123d 6092 . . . . . . 7 (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto→(Base‘)))
9 fveq2 6150 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
10 isismt.d . . . . . . . . . . . 12 𝐷 = (dist‘𝐺)
119, 10syl6eqr 2673 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
1211oveqd 6624 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
1312eqeq2d 2631 . . . . . . . . 9 (𝑔 = 𝐺 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
146, 13raleqbidv 3141 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
156, 14raleqbidv 3141 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
168, 15anbi12d 746 . . . . . 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 2738 . . . . 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 2622 . . . . . . . 8 ( = 𝐻𝑓 = 𝑓)
19 eqidd 2622 . . . . . . . 8 ( = 𝐻𝐵 = 𝐵)
20 fveq2 6150 . . . . . . . . 9 ( = 𝐻 → (Base‘) = (Base‘𝐻))
21 isismt.p . . . . . . . . 9 𝑃 = (Base‘𝐻)
2220, 21syl6eqr 2673 . . . . . . . 8 ( = 𝐻 → (Base‘) = 𝑃)
2318, 19, 22f1oeq123d 6092 . . . . . . 7 ( = 𝐻 → (𝑓:𝐵1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto𝑃))
24 fveq2 6150 . . . . . . . . . . 11 ( = 𝐻 → (dist‘) = (dist‘𝐻))
25 isismt.m . . . . . . . . . . 11 = (dist‘𝐻)
2624, 25syl6eqr 2673 . . . . . . . . . 10 ( = 𝐻 → (dist‘) = )
2726oveqd 6624 . . . . . . . . 9 ( = 𝐻 → ((𝑓𝑎)(dist‘)(𝑓𝑏)) = ((𝑓𝑎) (𝑓𝑏)))
2827eqeq1d 2623 . . . . . . . 8 ( = 𝐻 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
29282ralbidv 2983 . . . . . . 7 ( = 𝐻 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
3023, 29anbi12d 746 . . . . . 6 ( = 𝐻 → ((𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))))
3130abbidv 2738 . . . . 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 25335 . . . . 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 6635 . . . . . 6 (𝑃𝑚 𝐵) ∈ V
34 f1of 6096 . . . . . . . . 9 (𝑓:𝐵1-1-onto𝑃𝑓:𝐵𝑃)
35 fvex 6160 . . . . . . . . . . 11 (Base‘𝐻) ∈ V
3621, 35eqeltri 2694 . . . . . . . . . 10 𝑃 ∈ V
37 fvex 6160 . . . . . . . . . . 11 (Base‘𝐺) ∈ V
385, 37eqeltri 2694 . . . . . . . . . 10 𝐵 ∈ V
3936, 38elmap 7833 . . . . . . . . 9 (𝑓 ∈ (𝑃𝑚 𝐵) ↔ 𝑓:𝐵𝑃)
4034, 39sylibr 224 . . . . . . . 8 (𝑓:𝐵1-1-onto𝑃𝑓 ∈ (𝑃𝑚 𝐵))
4140adantr 481 . . . . . . 7 ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃𝑚 𝐵))
4241abssi 3658 . . . . . 6 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃𝑚 𝐵)
4333, 42ssexi 4765 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ∈ V
4417, 31, 32, 43ovmpt2 6752 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
451, 2, 44syl2an 494 . . 3 ((𝐺𝑉𝐻𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
4645eleq2d 2684 . 2 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))}))
47 f1of 6096 . . . . 5 (𝐹:𝐵1-1-onto𝑃𝐹:𝐵𝑃)
48 fex 6447 . . . . 5 ((𝐹:𝐵𝑃𝐵 ∈ V) → 𝐹 ∈ V)
4947, 38, 48sylancl 693 . . . 4 (𝐹:𝐵1-1-onto𝑃𝐹 ∈ V)
5049adantr 481 . . 3 ((𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
51 f1oeq1 6086 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐵1-1-onto𝑃𝐹:𝐵1-1-onto𝑃))
52 fveq1 6149 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑎) = (𝐹𝑎))
53 fveq1 6149 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑏) = (𝐹𝑏))
5452, 53oveq12d 6625 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑎) (𝑓𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
5554eqeq1d 2623 . . . . 5 (𝑓 = 𝐹 → (((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
56552ralbidv 2983 . . . 4 (𝑓 = 𝐹 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5751, 56anbi12d 746 . . 3 (𝑓 = 𝐹 → ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
5850, 57elab3 3342 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5946, 58syl6bb 276 1 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3186  wf 5845  1-1-ontowf1o 5848  cfv 5849  (class class class)co 6607  𝑚 cmap 7805  Basecbs 15784  distcds 15874  Ismtcismt 25334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-map 7807  df-ismt 25335
This theorem is referenced by:  ismot  25337
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