Theorem tgjustc2 26270

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) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 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.

Step	Hyp	Ref	Expression
1		tgjustc2.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2	1	fvexi 6659	. . . 4 ⊢ 𝑃 ∈ V
3	2, 2	xpex 7456	. . 3 ⊢ (𝑃 × 𝑃) ∈ V
4		tgjustc2.d	. . 3 ⊢ 𝑅 Er (𝑃 × 𝑃)
5		tgjustr 26268	. . 3 ⊢ (((𝑃 × 𝑃) ∈ V ∧ 𝑅 Er (𝑃 × 𝑃)) → ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣))))
6	3, 4, 5	mp2an 691	. 2 ⊢ ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)))
7		simplrl 776	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑤 ∈ 𝑃)
8		simplrr 777	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑥 ∈ 𝑃)
9	7, 8	opelxpd 5557	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃))
10		simprl 770	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑦 ∈ 𝑃)
11		simprr 772	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑧 ∈ 𝑃)
12	10, 11	opelxpd 5557	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃))
13		simpll 766	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)))
14		breq1 5033	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑅𝑣 ↔ ⟨𝑤, 𝑥⟩𝑅𝑣))
15		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑‘𝑢) = (𝑑‘⟨𝑤, 𝑥⟩))
16		df-ov 7138	. . . . . . . . . 10 ⊢ (𝑤𝑑𝑥) = (𝑑‘⟨𝑤, 𝑥⟩)
17	15, 16	eqtr4di 2851	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑‘𝑢) = (𝑤𝑑𝑥))
18	17	eqeq1d 2800	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑑‘𝑢) = (𝑑‘𝑣) ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣)))
19	14, 18	bibi12d 349	. . . . . . 7 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣))))
20		breq2 5034	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥⟩𝑅𝑣 ↔ ⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩))
21		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑‘𝑣) = (𝑑‘⟨𝑦, 𝑧⟩))
22		df-ov 7138	. . . . . . . . . 10 ⊢ (𝑦𝑑𝑧) = (𝑑‘⟨𝑦, 𝑧⟩)
23	21, 22	eqtr4di 2851	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑‘𝑣) = (𝑦𝑑𝑧))
24	23	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤𝑑𝑥) = (𝑑‘𝑣) ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
25	20, 24	bibi12d 349	. . . . . . 7 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥⟩𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
26	19, 25	rspc2va 3582	. . . . . 6 ⊢ (((⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃) ∧ ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃)) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣))) → (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
27	9, 12, 13, 26	syl21anc 836	. . . . 5 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
28	27	ralrimivva 3156	. . . 4 ⊢ ((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) → ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
29	28	ralrimivva 3156	. . 3 ⊢ (∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) → ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
30	29	anim2i 619	. 2 ⊢ ((𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣))) → (𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
31	6, 30	eximii 1838	1 ⊢ ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ⟨cop 4531   class class class wbr 5030   × cxp 5517   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   Er wer 8269  Basecbs 16475

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-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  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 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-er 8272  df-ec 8274  df-qs 8278

This theorem is referenced by: (None)