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Theorem tgjustc1 25826
 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
tgjustc1.p 𝑃 = (Base‘𝐺)
tgjustc1.d = (dist‘𝐺)
Assertion
Ref Expression
tgjustc1 𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
Distinct variable groups:   ,𝑟,𝑤,𝑥,𝑦,𝑧   𝑃,𝑟,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧,𝑤,𝑟)

Proof of Theorem tgjustc1
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgjustc1.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 tgjustf 25824 . . 3 ((𝑃 × 𝑃) ∈ V → ∃𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣))))
64, 5ax-mp 5 . 2 𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)))
7 simplrl 767 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑤𝑃)
8 simplrr 768 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑥𝑃)
97, 8opelxpd 5393 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃))
10 simprl 761 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑦𝑃)
11 simprr 763 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑧𝑃)
1210, 11opelxpd 5393 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃))
13 simpll 757 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)))
14 breq1 4889 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑟𝑣 ↔ ⟨𝑤, 𝑥𝑟𝑣))
15 fveq2 6446 . . . . . . . . . 10 (𝑢 = ⟨𝑤, 𝑥⟩ → ( 𝑢) = ( ‘⟨𝑤, 𝑥⟩))
16 df-ov 6925 . . . . . . . . . 10 (𝑤 𝑥) = ( ‘⟨𝑤, 𝑥⟩)
1715, 16syl6eqr 2832 . . . . . . . . 9 (𝑢 = ⟨𝑤, 𝑥⟩ → ( 𝑢) = (𝑤 𝑥))
1817eqeq1d 2780 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → (( 𝑢) = ( 𝑣) ↔ (𝑤 𝑥) = ( 𝑣)))
1914, 18bibi12d 337 . . . . . . 7 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ↔ (⟨𝑤, 𝑥𝑟𝑣 ↔ (𝑤 𝑥) = ( 𝑣))))
20 breq2 4890 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥𝑟𝑣 ↔ ⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩))
21 fveq2 6446 . . . . . . . . . 10 (𝑣 = ⟨𝑦, 𝑧⟩ → ( 𝑣) = ( ‘⟨𝑦, 𝑧⟩))
22 df-ov 6925 . . . . . . . . . 10 (𝑦 𝑧) = ( ‘⟨𝑦, 𝑧⟩)
2321, 22syl6eqr 2832 . . . . . . . . 9 (𝑣 = ⟨𝑦, 𝑧⟩ → ( 𝑣) = (𝑦 𝑧))
2423eqeq2d 2788 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤 𝑥) = ( 𝑣) ↔ (𝑤 𝑥) = (𝑦 𝑧)))
2520, 24bibi12d 337 . . . . . . 7 (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥𝑟𝑣 ↔ (𝑤 𝑥) = ( 𝑣)) ↔ (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧))))
2619, 25rspc2va 3525 . . . . . 6 (((⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃) ∧ ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃)) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣))) → (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
279, 12, 13, 26syl21anc 828 . . . . 5 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
2827ralrimivva 3153 . . . 4 ((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) → ∀𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
2928ralrimivva 3153 . . 3 (∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) → ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
3029anim2i 610 . 2 ((𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣))) → (𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧))))
316, 30eximii 1880 1 𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
 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  ‘cfv 6135  (class class class)co 6922   Er wer 8023  Basecbs 16255  distcds 16347 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-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-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  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-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fv 6143  df-ov 6925  df-er 8026 This theorem is referenced by: (None)
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