Theorem tgjustc2 28220

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 6896	. . . 4 ⊢ 𝑃 ∈ V
3	2, 2	xpex 7734	. . 3 ⊢ (𝑃 × 𝑃) ∈ V
4		tgjustc2.d	. . 3 ⊢ 𝑅 Er (𝑃 × 𝑃)
5		tgjustr 28218	. . 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 689	. 2 ⊢ ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)))
7		simplrl 774	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑤 ∈ 𝑃)
8		simplrr 775	. . . . . . 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 5706	. . . . . 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 768	. . . . . . 7 ⊢ (((∀𝑢 ∈ (𝑃 × 𝑃)∀𝑣 ∈ (𝑃 × 𝑃)(𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ∧ (𝑤 ∈ 𝑃 ∧ 𝑥 ∈ 𝑃)) ∧ (𝑦 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃)) → 𝑦 ∈ 𝑃)
11		simprr 770	. . . . . . 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 5706	. . . . . 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 764	. . . . . 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 5142	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑢𝑅𝑣 ↔ ⟨𝑤, 𝑥⟩𝑅𝑣))
15		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑‘𝑢) = (𝑑‘⟨𝑤, 𝑥⟩))
16		df-ov 7405	. . . . . . . . . 10 ⊢ (𝑤𝑑𝑥) = (𝑑‘⟨𝑤, 𝑥⟩)
17	15, 16	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → (𝑑‘𝑢) = (𝑤𝑑𝑥))
18	17	eqeq1d 2726	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑑‘𝑢) = (𝑑‘𝑣) ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣)))
19	14, 18	bibi12d 345	. . . . . . 7 ⊢ (𝑢 = ⟨𝑤, 𝑥⟩ → ((𝑢𝑅𝑣 ↔ (𝑑‘𝑢) = (𝑑‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣))))
20		breq2 5143	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (⟨𝑤, 𝑥⟩𝑅𝑣 ↔ ⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩))
21		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑‘𝑣) = (𝑑‘⟨𝑦, 𝑧⟩))
22		df-ov 7405	. . . . . . . . . 10 ⊢ (𝑦𝑑𝑧) = (𝑑‘⟨𝑦, 𝑧⟩)
23	21, 22	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → (𝑑‘𝑣) = (𝑦𝑑𝑧))
24	23	eqeq2d 2735	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((𝑤𝑑𝑥) = (𝑑‘𝑣) ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
25	20, 24	bibi12d 345	. . . . . . 7 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ((⟨𝑤, 𝑥⟩𝑅𝑣 ↔ (𝑤𝑑𝑥) = (𝑑‘𝑣)) ↔ (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))))
26	19, 25	rspc2va 3616	. . . . . 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 835	. . . . 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 3192	. . . 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 3192	. . 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 616	. 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 1831	1 ⊢ ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (⟨𝑤, 𝑥⟩𝑅⟨𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3053  Vcvv 3466  ⟨cop 4627   class class class wbr 5139   × cxp 5665   Fn wfn 6529  ‘cfv 6534  (class class class)co 7402   Er wer 8697  Basecbs 17149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-er 8700  df-ec 8702  df-qs 8706

This theorem is referenced by: (None)