Theorem tgjustc2 28938

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 6899	. . . 4 ⊢ 𝑃 ∈ V
3	2, 2	xpex 7767	. . 3 ⊢ (𝑃 × 𝑃) ∈ V
4		tgjustc2.d	. . 3 ⊢ 𝑅 Er (𝑃 × 𝑃)
5		tgjustr 28936	. . 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 705	. 2 ⊢ ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)))
7		simplrl 789	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑤 ∈ 𝑃)
8		simplrr 790	. . . . . . 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 5690	. . . . . 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 783	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑦 ∈ 𝑃)
11		simprr 785	. . . . . . 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 5690	. . . . . 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 779	. . . . . 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 5106	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑅𝑣 ↔ ⟨𝑤, 𝑥⟩𝑅𝑣))
15		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑‘𝑢) = (𝑑‘⟨𝑤, 𝑥⟩))
16		df-ov 7423	. . . . . . . . . 10 ⊢ (𝑤𝑑𝑥) = (𝑑‘⟨𝑤, 𝑥⟩)
17	15, 16	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑‘𝑢) = (𝑤𝑑𝑥))
18	17	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑑‘𝑢) = (𝑑‘𝑣) ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣)))
19	14, 18	bibi12d 348	. . . . . . 7 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣))))
20		breq2 5107	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥⟩𝑅𝑣 ↔ ⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩))
21		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑‘𝑣) = (𝑑‘⟨𝑦, 𝑧⟩))
22		df-ov 7423	. . . . . . . . . 10 ⊢ (𝑦𝑑𝑧) = (𝑑‘⟨𝑦, 𝑧⟩)
23	21, 22	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑‘𝑣) = (𝑦𝑑𝑧))
24	23	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤𝑑𝑥) = (𝑑‘𝑣) ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
25	20, 24	bibi12d 348	. . . . . . 7 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥⟩𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
26	19, 25	rspc2va 3588	. . . . . 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 851	. . . . 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 3206	. . . 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 3206	. . 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 629	. 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 1870	1 ⊢ ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   × cxp 5649   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420   Er wer 8714  Basecbs 17387

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-er 8717  df-ec 8719  df-qs 8723

This theorem is used by: (None)