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Theorem tgjustc1 26279
 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)
21fvexi 6660 . . . 4 𝑃 ∈ V
32, 2xpex 7459 . . 3 (𝑃 × 𝑃) ∈ V
4 tgjustf 26277 . . 3 ((𝑃 × 𝑃) ∈ V → ∃𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣))))
53, 4ax-mp 5 . 2 𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)))
6 simplrl 776 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑤𝑃)
7 simplrr 777 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑥𝑃)
86, 7opelxpd 5558 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃))
9 simprl 770 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑦𝑃)
10 simprr 772 . . . . . . 7 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → 𝑧𝑃)
119, 10opelxpd 5558 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃))
12 simpll 766 . . . . . 6 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)))
13 breq1 5034 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑟𝑣 ↔ ⟨𝑤, 𝑥𝑟𝑣))
14 fveq2 6646 . . . . . . . . . 10 (𝑢 = ⟨𝑤, 𝑥⟩ → ( 𝑢) = ( ‘⟨𝑤, 𝑥⟩))
15 df-ov 7139 . . . . . . . . . 10 (𝑤 𝑥) = ( ‘⟨𝑤, 𝑥⟩)
1614, 15eqtr4di 2851 . . . . . . . . 9 (𝑢 = ⟨𝑤, 𝑥⟩ → ( 𝑢) = (𝑤 𝑥))
1716eqeq1d 2800 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → (( 𝑢) = ( 𝑣) ↔ (𝑤 𝑥) = ( 𝑣)))
1813, 17bibi12d 349 . . . . . . 7 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ↔ (⟨𝑤, 𝑥𝑟𝑣 ↔ (𝑤 𝑥) = ( 𝑣))))
19 breq2 5035 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥𝑟𝑣 ↔ ⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩))
20 fveq2 6646 . . . . . . . . . 10 (𝑣 = ⟨𝑦, 𝑧⟩ → ( 𝑣) = ( ‘⟨𝑦, 𝑧⟩))
21 df-ov 7139 . . . . . . . . . 10 (𝑦 𝑧) = ( ‘⟨𝑦, 𝑧⟩)
2220, 21eqtr4di 2851 . . . . . . . . 9 (𝑣 = ⟨𝑦, 𝑧⟩ → ( 𝑣) = (𝑦 𝑧))
2322eqeq2d 2809 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤 𝑥) = ( 𝑣) ↔ (𝑤 𝑥) = (𝑦 𝑧)))
2419, 23bibi12d 349 . . . . . . 7 (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥𝑟𝑣 ↔ (𝑤 𝑥) = ( 𝑣)) ↔ (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧))))
2518, 24rspc2va 3582 . . . . . 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)))
268, 11, 12, 25syl21anc 836 . . . . 5 (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) ∧ (𝑦𝑃𝑧𝑃)) → (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
2726ralrimivva 3156 . . . 4 ((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) ∧ (𝑤𝑃𝑥𝑃)) → ∀𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
2827ralrimivva 3156 . . 3 (∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( 𝑢) = ( 𝑣)) → ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
2928anim2i 619 . 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))))
305, 29eximii 1838 1 𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ⟨cop 4531   class class class wbr 5031   × cxp 5518  ‘cfv 6325  (class class class)co 7136   Er wer 8272  Basecbs 16478  distcds 16569 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fv 6333  df-ov 7139  df-er 8275 This theorem is referenced by: (None)
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