MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isismt Structured version   Visualization version   GIF version

Theorem isismt 28557
Description: Property of being an isometry. Compare with isismty 37788. (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 3499 . . . 4 (𝐺𝑉𝐺 ∈ V)
2 elex 3499 . . . 4 (𝐻𝑊𝐻 ∈ V)
3 fveq2 6907 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 isismt.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2793 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65f1oeq2d 6845 . . . . . . 7 (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto→(Base‘)))
7 fveq2 6907 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8 isismt.d . . . . . . . . . . . 12 𝐷 = (dist‘𝐺)
97, 8eqtr4di 2793 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
109oveqd 7448 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
1110eqeq2d 2746 . . . . . . . . 9 (𝑔 = 𝐺 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
125, 11raleqbidv 3344 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
135, 12raleqbidv 3344 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
146, 13anbi12d 632 . . . . . 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))))
1514abbidv 2806 . . . . 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))})
16 fveq2 6907 . . . . . . . . 9 ( = 𝐻 → (Base‘) = (Base‘𝐻))
17 isismt.p . . . . . . . . 9 𝑃 = (Base‘𝐻)
1816, 17eqtr4di 2793 . . . . . . . 8 ( = 𝐻 → (Base‘) = 𝑃)
1918f1oeq3d 6846 . . . . . . 7 ( = 𝐻 → (𝑓:𝐵1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto𝑃))
20 fveq2 6907 . . . . . . . . . . 11 ( = 𝐻 → (dist‘) = (dist‘𝐻))
21 isismt.m . . . . . . . . . . 11 = (dist‘𝐻)
2220, 21eqtr4di 2793 . . . . . . . . . 10 ( = 𝐻 → (dist‘) = )
2322oveqd 7448 . . . . . . . . 9 ( = 𝐻 → ((𝑓𝑎)(dist‘)(𝑓𝑏)) = ((𝑓𝑎) (𝑓𝑏)))
2423eqeq1d 2737 . . . . . . . 8 ( = 𝐻 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
25242ralbidv 3219 . . . . . . 7 ( = 𝐻 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
2619, 25anbi12d 632 . . . . . 6 ( = 𝐻 → ((𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))))
2726abbidv 2806 . . . . 5 ( = 𝐻 → {𝑓 ∣ (𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏))} = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
28 df-ismt 28556 . . . . 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))})
29 ovex 7464 . . . . . 6 (𝑃m 𝐵) ∈ V
30 f1of 6849 . . . . . . . . 9 (𝑓:𝐵1-1-onto𝑃𝑓:𝐵𝑃)
3117fvexi 6921 . . . . . . . . . 10 𝑃 ∈ V
324fvexi 6921 . . . . . . . . . 10 𝐵 ∈ V
3331, 32elmap 8910 . . . . . . . . 9 (𝑓 ∈ (𝑃m 𝐵) ↔ 𝑓:𝐵𝑃)
3430, 33sylibr 234 . . . . . . . 8 (𝑓:𝐵1-1-onto𝑃𝑓 ∈ (𝑃m 𝐵))
3534adantr 480 . . . . . . 7 ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃m 𝐵))
3635abssi 4080 . . . . . 6 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃m 𝐵)
3729, 36ssexi 5328 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ∈ V
3815, 27, 28, 37ovmpo 7593 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
391, 2, 38syl2an 596 . . 3 ((𝐺𝑉𝐻𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
4039eleq2d 2825 . 2 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))}))
41 f1of 6849 . . . . 5 (𝐹:𝐵1-1-onto𝑃𝐹:𝐵𝑃)
42 fex 7246 . . . . 5 ((𝐹:𝐵𝑃𝐵 ∈ V) → 𝐹 ∈ V)
4341, 32, 42sylancl 586 . . . 4 (𝐹:𝐵1-1-onto𝑃𝐹 ∈ V)
4443adantr 480 . . 3 ((𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
45 f1oeq1 6837 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐵1-1-onto𝑃𝐹:𝐵1-1-onto𝑃))
46 fveq1 6906 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑎) = (𝐹𝑎))
47 fveq1 6906 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑏) = (𝐹𝑏))
4846, 47oveq12d 7449 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑎) (𝑓𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
4948eqeq1d 2737 . . . . 5 (𝑓 = 𝐹 → (((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
50492ralbidv 3219 . . . 4 (𝑓 = 𝐹 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5145, 50anbi12d 632 . . 3 (𝑓 = 𝐹 → ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
5244, 51elab3 3689 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5340, 52bitrdi 287 1 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  m cmap 8865  Basecbs 17245  distcds 17307  Ismtcismt 28555
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
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-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-iun 4998  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-ismt 28556
This theorem is referenced by:  ismot  28558
  Copyright terms: Public domain W3C validator