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Theorem isismt 26306
Description: Property of being an isometry. Compare with isismty 35119. (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 3489 . . . 4 (𝐺𝑉𝐺 ∈ V)
2 elex 3489 . . . 4 (𝐻𝑊𝐻 ∈ V)
3 fveq2 6643 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 isismt.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
53, 4syl6eqr 2874 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65f1oeq2d 6584 . . . . . . 7 (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto→(Base‘)))
7 fveq2 6643 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8 isismt.d . . . . . . . . . . . 12 𝐷 = (dist‘𝐺)
97, 8syl6eqr 2874 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
109oveqd 7147 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
1110eqeq2d 2832 . . . . . . . . 9 (𝑔 = 𝐺 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
125, 11raleqbidv 3386 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
135, 12raleqbidv 3386 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)))
146, 13anbi12d 633 . . . . . 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 2885 . . . . 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 6643 . . . . . . . . 9 ( = 𝐻 → (Base‘) = (Base‘𝐻))
17 isismt.p . . . . . . . . 9 𝑃 = (Base‘𝐻)
1816, 17syl6eqr 2874 . . . . . . . 8 ( = 𝐻 → (Base‘) = 𝑃)
1918f1oeq3d 6585 . . . . . . 7 ( = 𝐻 → (𝑓:𝐵1-1-onto→(Base‘) ↔ 𝑓:𝐵1-1-onto𝑃))
20 fveq2 6643 . . . . . . . . . . 11 ( = 𝐻 → (dist‘) = (dist‘𝐻))
21 isismt.m . . . . . . . . . . 11 = (dist‘𝐻)
2220, 21syl6eqr 2874 . . . . . . . . . 10 ( = 𝐻 → (dist‘) = )
2322oveqd 7147 . . . . . . . . 9 ( = 𝐻 → ((𝑓𝑎)(dist‘)(𝑓𝑏)) = ((𝑓𝑎) (𝑓𝑏)))
2423eqeq1d 2823 . . . . . . . 8 ( = 𝐻 → (((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
25242ralbidv 3187 . . . . . . 7 ( = 𝐻 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)))
2619, 25anbi12d 633 . . . . . 6 ( = 𝐻 → ((𝑓:𝐵1-1-onto→(Base‘) ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))))
2726abbidv 2885 . . . . 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 26305 . . . . 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 7163 . . . . . 6 (𝑃m 𝐵) ∈ V
30 f1of 6588 . . . . . . . . 9 (𝑓:𝐵1-1-onto𝑃𝑓:𝐵𝑃)
3117fvexi 6657 . . . . . . . . . 10 𝑃 ∈ V
324fvexi 6657 . . . . . . . . . 10 𝐵 ∈ V
3331, 32elmap 8410 . . . . . . . . 9 (𝑓 ∈ (𝑃m 𝐵) ↔ 𝑓:𝐵𝑃)
3430, 33sylibr 237 . . . . . . . 8 (𝑓:𝐵1-1-onto𝑃𝑓 ∈ (𝑃m 𝐵))
3534adantr 484 . . . . . . 7 ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃m 𝐵))
3635abssi 4022 . . . . . 6 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃m 𝐵)
3729, 36ssexi 5199 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ∈ V
3815, 27, 28, 37ovmpo 7284 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
391, 2, 38syl2an 598 . . 3 ((𝐺𝑉𝐻𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))})
4039eleq2d 2897 . 2 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))}))
41 f1of 6588 . . . . 5 (𝐹:𝐵1-1-onto𝑃𝐹:𝐵𝑃)
42 fex 6962 . . . . 5 ((𝐹:𝐵𝑃𝐵 ∈ V) → 𝐹 ∈ V)
4341, 32, 42sylancl 589 . . . 4 (𝐹:𝐵1-1-onto𝑃𝐹 ∈ V)
4443adantr 484 . . 3 ((𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
45 f1oeq1 6577 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐵1-1-onto𝑃𝐹:𝐵1-1-onto𝑃))
46 fveq1 6642 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑎) = (𝐹𝑎))
47 fveq1 6642 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑏) = (𝐹𝑏))
4846, 47oveq12d 7148 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑎) (𝑓𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
4948eqeq1d 2823 . . . . 5 (𝑓 = 𝐹 → (((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
50492ralbidv 3187 . . . 4 (𝑓 = 𝐹 → (∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5145, 50anbi12d 633 . . 3 (𝑓 = 𝐹 → ((𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏)) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
5244, 51elab3 3651 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑓𝑎) (𝑓𝑏)) = (𝑎𝐷𝑏))} ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏)))
5340, 52syl6bb 290 1 ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2799  wral 3126  Vcvv 3471  wf 6324  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7130  m cmap 8381  Basecbs 16461  distcds 16552  Ismtcismt 26304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-ismt 26305
This theorem is referenced by:  ismot  26307
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