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Theorem tgjustc1 27993
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 = (Baseβ€˜πΊ)
tgjustc1.d βˆ’ = (distβ€˜πΊ)
Assertion
Ref Expression
tgjustc1 βˆƒπ‘Ÿ(π‘Ÿ Er (𝑃 Γ— 𝑃) ∧ βˆ€π‘€ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
Distinct variable groups:   βˆ’ ,π‘Ÿ,𝑀,π‘₯,𝑦,𝑧   𝑃,π‘Ÿ,𝑀,π‘₯,𝑦,𝑧
Allowed substitution hints:   𝐺(π‘₯,𝑦,𝑧,𝑀,π‘Ÿ)

Proof of Theorem tgjustc1
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgjustc1.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
21fvexi 6904 . . . 4 𝑃 ∈ V
32, 2xpex 7742 . . 3 (𝑃 Γ— 𝑃) ∈ V
4 tgjustf 27991 . . 3 ((𝑃 Γ— 𝑃) ∈ V β†’ βˆƒπ‘Ÿ(π‘Ÿ Er (𝑃 Γ— 𝑃) ∧ βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£))))
53, 4ax-mp 5 . 2 βˆƒπ‘Ÿ(π‘Ÿ Er (𝑃 Γ— 𝑃) ∧ βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)))
6 simplrl 773 . . . . . . 7 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ 𝑀 ∈ 𝑃)
7 simplrr 774 . . . . . . 7 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ π‘₯ ∈ 𝑃)
86, 7opelxpd 5714 . . . . . 6 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ βŸ¨π‘€, π‘₯⟩ ∈ (𝑃 Γ— 𝑃))
9 simprl 767 . . . . . . 7 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ 𝑦 ∈ 𝑃)
10 simprr 769 . . . . . . 7 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ 𝑧 ∈ 𝑃)
119, 10opelxpd 5714 . . . . . 6 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ βŸ¨π‘¦, π‘§βŸ© ∈ (𝑃 Γ— 𝑃))
12 simpll 763 . . . . . 6 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)))
13 breq1 5150 . . . . . . . 8 (𝑒 = βŸ¨π‘€, π‘₯⟩ β†’ (π‘’π‘Ÿπ‘£ ↔ βŸ¨π‘€, π‘₯βŸ©π‘Ÿπ‘£))
14 fveq2 6890 . . . . . . . . . 10 (𝑒 = βŸ¨π‘€, π‘₯⟩ β†’ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜βŸ¨π‘€, π‘₯⟩))
15 df-ov 7414 . . . . . . . . . 10 (𝑀 βˆ’ π‘₯) = ( βˆ’ β€˜βŸ¨π‘€, π‘₯⟩)
1614, 15eqtr4di 2788 . . . . . . . . 9 (𝑒 = βŸ¨π‘€, π‘₯⟩ β†’ ( βˆ’ β€˜π‘’) = (𝑀 βˆ’ π‘₯))
1716eqeq1d 2732 . . . . . . . 8 (𝑒 = βŸ¨π‘€, π‘₯⟩ β†’ (( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£) ↔ (𝑀 βˆ’ π‘₯) = ( βˆ’ β€˜π‘£)))
1813, 17bibi12d 344 . . . . . . 7 (𝑒 = βŸ¨π‘€, π‘₯⟩ β†’ ((π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ↔ (βŸ¨π‘€, π‘₯βŸ©π‘Ÿπ‘£ ↔ (𝑀 βˆ’ π‘₯) = ( βˆ’ β€˜π‘£))))
19 breq2 5151 . . . . . . . 8 (𝑣 = βŸ¨π‘¦, π‘§βŸ© β†’ (βŸ¨π‘€, π‘₯βŸ©π‘Ÿπ‘£ ↔ βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ©))
20 fveq2 6890 . . . . . . . . . 10 (𝑣 = βŸ¨π‘¦, π‘§βŸ© β†’ ( βˆ’ β€˜π‘£) = ( βˆ’ β€˜βŸ¨π‘¦, π‘§βŸ©))
21 df-ov 7414 . . . . . . . . . 10 (𝑦 βˆ’ 𝑧) = ( βˆ’ β€˜βŸ¨π‘¦, π‘§βŸ©)
2220, 21eqtr4di 2788 . . . . . . . . 9 (𝑣 = βŸ¨π‘¦, π‘§βŸ© β†’ ( βˆ’ β€˜π‘£) = (𝑦 βˆ’ 𝑧))
2322eqeq2d 2741 . . . . . . . 8 (𝑣 = βŸ¨π‘¦, π‘§βŸ© β†’ ((𝑀 βˆ’ π‘₯) = ( βˆ’ β€˜π‘£) ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
2419, 23bibi12d 344 . . . . . . 7 (𝑣 = βŸ¨π‘¦, π‘§βŸ© β†’ ((βŸ¨π‘€, π‘₯βŸ©π‘Ÿπ‘£ ↔ (𝑀 βˆ’ π‘₯) = ( βˆ’ β€˜π‘£)) ↔ (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧))))
2518, 24rspc2va 3622 . . . . . 6 (((βŸ¨π‘€, π‘₯⟩ ∈ (𝑃 Γ— 𝑃) ∧ βŸ¨π‘¦, π‘§βŸ© ∈ (𝑃 Γ— 𝑃)) ∧ βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£))) β†’ (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
268, 11, 12, 25syl21anc 834 . . . . 5 (((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) β†’ (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
2726ralrimivva 3198 . . . 4 ((βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) ∧ (𝑀 ∈ 𝑃 ∧ π‘₯ ∈ 𝑃)) β†’ βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
2827ralrimivva 3198 . . 3 (βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£)) β†’ βˆ€π‘€ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
2928anim2i 615 . 2 ((π‘Ÿ Er (𝑃 Γ— 𝑃) ∧ βˆ€π‘’ ∈ (𝑃 Γ— 𝑃)βˆ€π‘£ ∈ (𝑃 Γ— 𝑃)(π‘’π‘Ÿπ‘£ ↔ ( βˆ’ β€˜π‘’) = ( βˆ’ β€˜π‘£))) β†’ (π‘Ÿ Er (𝑃 Γ— 𝑃) ∧ βˆ€π‘€ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧))))
305, 29eximii 1837 1 βˆƒπ‘Ÿ(π‘Ÿ Er (𝑃 Γ— 𝑃) ∧ βˆ€π‘€ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472  βŸ¨cop 4633   class class class wbr 5147   Γ— cxp 5673  β€˜cfv 6542  (class class class)co 7411   Er wer 8702  Basecbs 17148  distcds 17210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fv 6550  df-ov 7414  df-er 8705
This theorem is referenced by: (None)
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