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Theorem tgjustc2 27467
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 6860 . . . 4 𝑃 ∈ V
32, 2xpex 7691 . . 3 (𝑃 × 𝑃) ∈ V
4 tgjustc2.d . . 3 𝑅 Er (𝑃 × 𝑃)
5 tgjustr 27465 . . 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 5675 . . . . . 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 5675 . . . . . 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 5112 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑅𝑣 ↔ ⟨𝑤, 𝑥𝑅𝑣))
15 fveq2 6846 . . . . . . . . . 10 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑𝑢) = (𝑑‘⟨𝑤, 𝑥⟩))
16 df-ov 7364 . . . . . . . . . 10 (𝑤𝑑𝑥) = (𝑑‘⟨𝑤, 𝑥⟩)
1715, 16eqtr4di 2791 . . . . . . . . 9 (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑𝑢) = (𝑤𝑑𝑥))
1817eqeq1d 2735 . . . . . . . 8 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑑𝑢) = (𝑑𝑣) ↔ (𝑤𝑑𝑥) = (𝑑𝑣)))
1914, 18bibi12d 346 . . . . . . 7 (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ↔ (⟨𝑤, 𝑥𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑𝑣))))
20 breq2 5113 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥𝑅𝑣 ↔ ⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩))
21 fveq2 6846 . . . . . . . . . 10 (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑𝑣) = (𝑑‘⟨𝑦, 𝑧⟩))
22 df-ov 7364 . . . . . . . . . 10 (𝑦𝑑𝑧) = (𝑑‘⟨𝑦, 𝑧⟩)
2321, 22eqtr4di 2791 . . . . . . . . 9 (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑𝑣) = (𝑦𝑑𝑧))
2423eqeq2d 2744 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤𝑑𝑥) = (𝑑𝑣) ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2520, 24bibi12d 346 . . . . . . 7 (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑𝑣)) ↔ (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
2619, 25rspc2va 3593 . . . . . 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 3194 . . . 4 ((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) ∧ (𝑤𝑃𝑥𝑃)) → ∀𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
2928ralrimivva 3194 . . 3 (∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑𝑢) = (𝑑𝑣)) → ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
3029anim2i 618 . 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 1840 1 𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3061  Vcvv 3447  cop 4596   class class class wbr 5109   × cxp 5635   Fn wfn 6495  cfv 6500  (class class class)co 7361   Er wer 8651  Basecbs 17091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-er 8654  df-ec 8656  df-qs 8660
This theorem is referenced by: (None)
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