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Theorem tgjustc2 28279
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)
21fvexi 6911 . . . 4 𝑃 ∈ V
32, 2xpex 7755 . . 3 (𝑃 × 𝑃) ∈ V
4 tgjustc2.d . . 3 𝑅 Er (𝑃 × 𝑃)
5 tgjustr 28277 . . 3 (((𝑃 × 𝑃) ∈ V ∧ 𝑅 Er (𝑃 × 𝑃)) → ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣))))
63, 4, 5mp2an 691 . 2 𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)))
7 simplrl 776 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑤𝑃)
8 simplrr 777 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑥𝑃)
97, 8opelxpd 5717 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃))
10 simprl 770 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑦𝑃)
11 simprr 772 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑧𝑃)
1210, 11opelxpd 5717 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃))
13 simpll 766 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)))
14 breq1 5151 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑅𝑣 ↔ ⟨𝑤, 𝑥𝑅𝑣))
15 fveq2 6897 . . . . . . . . . 10 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑𝑢) = (𝑑‘⟨𝑤, 𝑥⟩))
16 df-ov 7423 . . . . . . . . . 10 (𝑤𝑑𝑥) = (𝑑‘⟨𝑤, 𝑥⟩)
1715, 16eqtr4di 2786 . . . . . . . . 9 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑𝑢) = (𝑤𝑑𝑥))
1817eqeq1d 2730 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑑𝑢) = (𝑑𝑣) ↔ (𝑤𝑑𝑥) = (𝑑𝑣)))
1914, 18bibi12d 345 . . . . . . 7 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ↔ (⟨𝑤, 𝑥𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑𝑣))))
20 breq2 5152 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥𝑅𝑣 ↔ ⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩))
21 fveq2 6897 . . . . . . . . . 10 (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑𝑣) = (𝑑‘⟨𝑦, 𝑧⟩))
22 df-ov 7423 . . . . . . . . . 10 (𝑦𝑑𝑧) = (𝑑‘⟨𝑦, 𝑧⟩)
2321, 22eqtr4di 2786 . . . . . . . . 9 (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑𝑣) = (𝑦𝑑𝑧))
2423eqeq2d 2739 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤𝑑𝑥) = (𝑑𝑣) ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2520, 24bibi12d 345 . . . . . . 7 (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑𝑣)) ↔ (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
2619, 25rspc2va 3621 . . . . . 6 (((⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃) ∧ ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃)) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣))) → (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
279, 12, 13, 26syl21anc 837 . . . . 5 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2827ralrimivva 3197 . . . 4 ((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) → ∀𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2928ralrimivva 3197 . . 3 (∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) → ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
3029anim2i 616 . 2 ((𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣))) → (𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
316, 30eximii 1832 1 𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wex 1774  wcel 2099  wral 3058  Vcvv 3471  cop 4635   class class class wbr 5148   × cxp 5676   Fn wfn 6543  cfv 6548  (class class class)co 7420   Er wer 8721  Basecbs 17179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-er 8724  df-ec 8726  df-qs 8730
This theorem is referenced by: (None)
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