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Theorem isismt 25229
 Description: Property of being an isometry. Compare with isismty 32770. (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 3185 . . . 4 (𝐺𝑉𝐺 ∈ V)
2 elex 3185 . . . 4 (𝐻𝑊𝐻 ∈ V)
3 eqidd 2611 . . . . . . . 8 (𝑔 = 𝐺𝑓 = 𝑓)
4 fveq2 6103 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 isismt.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
64, 5syl6eqr 2662 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7 eqidd 2611 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘) = (Base‘))
83, 6, 7f1oeq123d 6046 . . . . . . 7 (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto→(Base‘)))
9 fveq2 6103 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
10 isismt.d . . . . . . . . . . . 12 𝐷 = (dist‘𝐺)
119, 10syl6eqr 2662 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
1211oveqd 6566 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
1312eqeq2d 2620 . . . . . . . . 9 (𝑔 = 𝐺 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
146, 13raleqbidv 3129 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
156, 14raleqbidv 3129 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
168, 15anbi12d 743 . . . . . 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 2728 . . . . 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 2611 . . . . . . . 8 ( = 𝐻𝑓 = 𝑓)
19 eqidd 2611 . . . . . . . 8 ( = 𝐻𝐵 = 𝐵)
20 fveq2 6103 . . . . . . . . 9 ( = 𝐻 → (Base‘) = (Base‘𝐻))
21 isismt.p . . . . . . . . 9 𝑃 = (Base‘𝐻)
2220, 21syl6eqr 2662 . . . . . . . 8 ( = 𝐻 → (Base‘) = 𝑃)
2318, 19, 22f1oeq123d 6046 . . . . . . 7 ( = 𝐻 → (𝑓:𝐵1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto𝑃))
24 fveq2 6103 . . . . . . . . . . 11 ( = 𝐻 → (dist‘) = (dist‘𝐻))
25 isismt.m . . . . . . . . . . 11 = (dist‘𝐻)
2624, 25syl6eqr 2662 . . . . . . . . . 10 ( = 𝐻 → (dist‘) = )
2726oveqd 6566 . . . . . . . . 9 ( = 𝐻 → ((𝑓𝑎)(dist‘)(𝑓𝑏)) = ((𝑓𝑎) (𝑓𝑏)))
2827eqeq1d 2612 . . . . . . . 8 ( = 𝐻 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
29282ralbidv 2972 . . . . . . 7 ( = 𝐻 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
3023, 29anbi12d 743 . . . . . 6 ( = 𝐻 → ((𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))))
3130abbidv 2728 . . . . 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 25228 . . . . 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 6577 . . . . . 6 (𝑃𝑚 𝐵) ∈ V
34 f1of 6050 . . . . . . . . 9 (𝑓:𝐵1-1-onto𝑃𝑓:𝐵𝑃)
35 fvex 6113 . . . . . . . . . . 11 (Base‘𝐻) ∈ V
3621, 35eqeltri 2684 . . . . . . . . . 10 𝑃 ∈ V
37 fvex 6113 . . . . . . . . . . 11 (Base‘𝐺) ∈ V
385, 37eqeltri 2684 . . . . . . . . . 10 𝐵 ∈ V
3936, 38elmap 7772 . . . . . . . . 9 (𝑓 ∈ (𝑃𝑚 𝐵) ↔ 𝑓:𝐵𝑃)
4034, 39sylibr 223 . . . . . . . 8 (𝑓:𝐵1-1-onto𝑃𝑓 ∈ (𝑃𝑚 𝐵))
4140adantr 480 . . . . . . 7 ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃𝑚 𝐵))
4241abssi 3640 . . . . . 6 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃𝑚 𝐵)
4333, 42ssexi 4731 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ∈ V
4417, 31, 32, 43ovmpt2 6694 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
451, 2, 44syl2an 493 . . 3 ((𝐺𝑉𝐻𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
4645eleq2d 2673 . 2 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))}))
47 f1of 6050 . . . . 5 (𝐹:𝐵1-1-onto𝑃𝐹:𝐵𝑃)
48 fex 6394 . . . . 5 ((𝐹:𝐵𝑃𝐵 ∈ V) → 𝐹 ∈ V)
4947, 38, 48sylancl 693 . . . 4 (𝐹:𝐵1-1-onto𝑃𝐹 ∈ V)
5049adantr 480 . . 3 ((𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
51 f1oeq1 6040 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐵1-1-onto𝑃𝐹:𝐵1-1-onto𝑃))
52 fveq1 6102 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑎) = (𝐹𝑎))
53 fveq1 6102 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑏) = (𝐹𝑏))
5452, 53oveq12d 6567 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑎) (𝑓𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
5554eqeq1d 2612 . . . . 5 (𝑓 = 𝐹 → (((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
56552ralbidv 2972 . . . 4 (𝑓 = 𝐹 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5751, 56anbi12d 743 . . 3 (𝑓 = 𝐹 → ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
5850, 57elab3 3327 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5946, 58syl6bb 275 1 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Basecbs 15695  distcds 15777  Ismtcismt 25227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ismt 25228 This theorem is referenced by:  ismot  25230
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