Theorem tgjustc1 28952

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

Step	Hyp	Ref	Expression
1		tgjustc1.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2	1	fvexi 6910	. . . 4 ⊢ 𝑃 ∈ V
3	2, 2	xpex 7777	. . 3 ⊢ (𝑃 × 𝑃) ∈ V
4		tgjustf 28950	. . 3 ⊢ ((𝑃 × 𝑃) ∈ V → ∃𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣))))
5	3, 4	ax-mp 5	. 2 ⊢ ∃𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)))
6		simplrl 789	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑤 ∈ 𝑃)
7		simplrr 790	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑥 ∈ 𝑃)
8	6, 7	opelxpd 5694	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃))
9		simprl 783	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑦 ∈ 𝑃)
10		simprr 785	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑧 ∈ 𝑃)
11	9, 10	opelxpd 5694	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃))
12		simpll 779	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)))
13		breq1 5109	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑟𝑣 ↔ ⟨𝑤, 𝑥⟩𝑟𝑣))
14		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ( − ‘𝑢) = ( − ‘⟨𝑤, 𝑥⟩))
15		df-ov 7433	. . . . . . . . . 10 ⊢ (𝑤 − 𝑥) = ( − ‘⟨𝑤, 𝑥⟩)
16	14, 15	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ( − ‘𝑢) = (𝑤 − 𝑥))
17	16	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (( − ‘𝑢) = ( − ‘𝑣) ↔ (𝑤 − 𝑥) = ( − ‘𝑣)))
18	13, 17	bibi12d 348	. . . . . . 7 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑟𝑣 ↔ (𝑤 − 𝑥) = ( − ‘𝑣))))
19		breq2 5110	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥⟩𝑟𝑣 ↔ ⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩))
20		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ( − ‘𝑣) = ( − ‘⟨𝑦, 𝑧⟩))
21		df-ov 7433	. . . . . . . . . 10 ⊢ (𝑦 − 𝑧) = ( − ‘⟨𝑦, 𝑧⟩)
22	20, 21	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ( − ‘𝑣) = (𝑦 − 𝑧))
23	22	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤 − 𝑥) = ( − ‘𝑣) ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧)))
24	19, 23	bibi12d 348	. . . . . . 7 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥⟩𝑟𝑣 ↔ (𝑤 − 𝑥) = ( − ‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧))))
25	18, 24	rspc2va 3591	. . . . . 6 ⊢ (((⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃) ∧ ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃)) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣))) → (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧)))
26	8, 11, 12, 25	syl21anc 851	. . . . 5 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧)))
27	26	ralrimivva 3209	. . . 4 ⊢ ((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) → ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧)))
28	27	ralrimivva 3209	. . 3 ⊢ (∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) → ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧)))
29	28	anim2i 629	. 2 ⊢ ((𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣))) → (𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧))))
30	5, 29	eximii 1869	1 ⊢ ∃𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∀wral 3080  Vcvv 3454  ⟨cop 4838   class class class wbr 5106   × cxp 5653  ‘cfv 6547  (class class class)co 7430   Er wer 8728  Basecbs 17402  distcds 17452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6501  df-fv 6555  df-ov 7433  df-er 8731

This theorem is used by: (None)