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Theorem ltgseg 27712
Description: The set 𝐸 denotes the possible values of the congruence. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Hypotheses
Ref Expression
legval.p 𝑃 = (Base‘𝐺)
legval.d = (dist‘𝐺)
legval.i 𝐼 = (Itv‘𝐺)
legval.l = (≤G‘𝐺)
legval.g (𝜑𝐺 ∈ TarskiG)
legso.a 𝐸 = ( “ (𝑃 × 𝑃))
legso.f (𝜑 → Fun )
ltgseg.p (𝜑𝐴𝐸)
Assertion
Ref Expression
ltgseg (𝜑 → ∃𝑥𝑃𝑦𝑃 𝐴 = (𝑥 𝑦))
Distinct variable groups:   𝑥, ,𝑦   𝑥,𝐴,𝑦   𝑥,𝑃,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐸(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐼(𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem ltgseg
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simp-4r 782 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = 𝐴)
2 simpr 485 . . . . . 6 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
32fveq2d 6882 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = ( ‘⟨𝑥, 𝑦⟩))
41, 3eqtr3d 2773 . . . 4 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( ‘⟨𝑥, 𝑦⟩))
5 df-ov 7396 . . . 4 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
64, 5eqtr4di 2789 . . 3 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 𝑦))
7 simplr 767 . . . 4 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8 elxp2 5693 . . . 4 (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
97, 8sylib 217 . . 3 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
106, 9reximddv2 3211 . 2 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → ∃𝑥𝑃𝑦𝑃 𝐴 = (𝑥 𝑦))
11 legso.f . . 3 (𝜑 → Fun )
12 ltgseg.p . . . 4 (𝜑𝐴𝐸)
13 legso.a . . . 4 𝐸 = ( “ (𝑃 × 𝑃))
1412, 13eleqtrdi 2842 . . 3 (𝜑𝐴 ∈ ( “ (𝑃 × 𝑃)))
15 fvelima 6944 . . 3 ((Fun 𝐴 ∈ ( “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1611, 14, 15syl2anc 584 . 2 (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1710, 16r19.29a 3161 1 (𝜑 → ∃𝑥𝑃𝑦𝑃 𝐴 = (𝑥 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3069  cop 4628   × cxp 5667  cima 5672  Fun wfun 6526  cfv 6532  (class class class)co 7393  Basecbs 17126  distcds 17188  TarskiGcstrkg 27543  Itvcitv 27549  ≤Gcleg 27698
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fv 6540  df-ov 7396
This theorem is referenced by:  legso  27715
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