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Theorem tgjustc2 25827
 Description: A justification for using distance equality instead of the textbook relation on pairs of points for congruence. (Contributed by Thierry Arnoux, 29-Jan-2023.)
Hypotheses
Ref Expression
tgjustc2.p 𝑃 = (Base‘𝐺)
tgjustc2.d 𝑅 Er (𝑃 × 𝑃)
Assertion
Ref Expression
tgjustc2 𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
Distinct variable groups:   𝑃,𝑑,𝑤,𝑥,𝑦,𝑧   𝑅,𝑑,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧,𝑤,𝑑)

Proof of Theorem tgjustc2
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgjustc2.p . . . . 5 𝑃 = (Base‘𝐺)
2 fvex 6459 . . . . 5 (Base‘𝐺) ∈ V
31, 2eqeltri 2855 . . . 4 𝑃 ∈ V
43, 3xpex 7240 . . 3 (𝑃 × 𝑃) ∈ V
5 tgjustc2.d . . 3 𝑅 Er (𝑃 × 𝑃)
6 tgjustr 25825 . . 3 (((𝑃 × 𝑃) ∈ V ∧ 𝑅 Er (𝑃 × 𝑃)) → ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣))))
74, 5, 6mp2an 682 . 2 𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)))
8 simplrl 767 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑤𝑃)
9 simplrr 768 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑥𝑃)
108, 9opelxpd 5393 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃))
11 simprl 761 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑦𝑃)
12 simprr 763 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑧𝑃)
1311, 12opelxpd 5393 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃))
14 simpll 757 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)))
15 breq1 4889 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑅𝑣 ↔ ⟨𝑤, 𝑥𝑅𝑣))
16 fveq2 6446 . . . . . . . . . 10 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑𝑢) = (𝑑‘⟨𝑤, 𝑥⟩))
17 df-ov 6925 . . . . . . . . . 10 (𝑤𝑑𝑥) = (𝑑‘⟨𝑤, 𝑥⟩)
1816, 17syl6eqr 2832 . . . . . . . . 9 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑𝑢) = (𝑤𝑑𝑥))
1918eqeq1d 2780 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑑𝑢) = (𝑑𝑣) ↔ (𝑤𝑑𝑥) = (𝑑𝑣)))
2015, 19bibi12d 337 . . . . . . 7 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ↔ (⟨𝑤, 𝑥𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑𝑣))))
21 breq2 4890 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥𝑅𝑣 ↔ ⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩))
22 fveq2 6446 . . . . . . . . . 10 (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑𝑣) = (𝑑‘⟨𝑦, 𝑧⟩))
23 df-ov 6925 . . . . . . . . . 10 (𝑦𝑑𝑧) = (𝑑‘⟨𝑦, 𝑧⟩)
2422, 23syl6eqr 2832 . . . . . . . . 9 (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑𝑣) = (𝑦𝑑𝑧))
2524eqeq2d 2788 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤𝑑𝑥) = (𝑑𝑣) ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2621, 25bibi12d 337 . . . . . . 7 (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑𝑣)) ↔ (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
2720, 26rspc2va 3525 . . . . . 6 (((⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃) ∧ ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃)) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣))) → (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2810, 13, 14, 27syl21anc 828 . . . . 5 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2928ralrimivva 3153 . . . 4 ((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) → ∀𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
3029ralrimivva 3153 . . 3 (∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) → ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
3130anim2i 610 . 2 ((𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣))) → (𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
327, 31eximii 1880 1 𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∀wral 3090  Vcvv 3398  ⟨cop 4404   class class class wbr 4886   × cxp 5353   Fn wfn 6130  ‘cfv 6135  (class class class)co 6922   Er wer 8023  Basecbs 16255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-er 8026  df-ec 8028  df-qs 8032 This theorem is referenced by: (None)
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