Theorem tgjustc1 26259

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 6677	. . . 4 ⊢ 𝑃 ∈ V
3	2, 2	xpex 7469	. . 3 ⊢ (𝑃 × 𝑃) ∈ V
4		tgjustf 26257	. . 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 775	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑤 ∈ 𝑃)
7		simplrr 776	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑥 ∈ 𝑃)
8	6, 7	opelxpd 5586	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ⟨𝑤, 𝑥⟩ ∈ (𝑃 × 𝑃))
9		simprl 769	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑦 ∈ 𝑃)
10		simprr 771	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑧 ∈ 𝑃)
11	9, 10	opelxpd 5586	. . . . . 6 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑟𝑣 ↔ ( − ‘𝑢) = ( − ‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → ⟨𝑦, 𝑧⟩ ∈ (𝑃 × 𝑃))
12		simpll 765	. . . . . 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 5062	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑟𝑣 ↔ ⟨𝑤, 𝑥⟩𝑟𝑣))
14		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ( − ‘𝑢) = ( − ‘⟨𝑤, 𝑥⟩))
15		df-ov 7152	. . . . . . . . . 10 ⊢ (𝑤 − 𝑥) = ( − ‘⟨𝑤, 𝑥⟩)
16	14, 15	syl6eqr 2873	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ( − ‘𝑢) = (𝑤 − 𝑥))
17	16	eqeq1d 2822	. . . . . . . 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 5063	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥⟩𝑟𝑣 ↔ ⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩))
20		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ( − ‘𝑣) = ( − ‘⟨𝑦, 𝑧⟩))
21		df-ov 7152	. . . . . . . . . 10 ⊢ (𝑦 − 𝑧) = ( − ‘⟨𝑦, 𝑧⟩)
22	20, 21	syl6eqr 2873	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ( − ‘𝑣) = (𝑦 − 𝑧))
23	22	eqeq2d 2831	. . . . . . . 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 3631	. . . . . 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 835	. . . . 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 3190	. . . 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 3190	. . 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 618	. 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 1836	1 ⊢ ∃𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑟⟨𝑦, 𝑧⟩ ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧)))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113  ∀wral 3137  Vcvv 3491  ⟨cop 4566   class class class wbr 5059   × cxp 5546  ‘cfv 6348  (class class class)co 7149   Er wer 8279  Basecbs 16478  distcds 16569

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7152  df-er 8282

This theorem is referenced by: (None)