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Theorem ltgseg 25291
 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 803 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = 𝐴)
2 simpr 476 . . . . . 6 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
32fveq2d 6107 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = ( ‘⟨𝑥, 𝑦⟩))
41, 3eqtr3d 2646 . . . 4 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( ‘⟨𝑥, 𝑦⟩))
5 df-ov 6552 . . . 4 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
64, 5syl6eqr 2662 . . 3 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 𝑦))
7 simplr 788 . . . 4 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8 elxp2 5056 . . . 4 (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
97, 8sylib 207 . . 3 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
106, 9reximddv2 3002 . 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, 13syl6eleq 2698 . . 3 (𝜑𝐴 ∈ ( “ (𝑃 × 𝑃)))
15 fvelima 6158 . . 3 ((Fun 𝐴 ∈ ( “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1611, 14, 15syl2anc 691 . 2 (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1710, 16r19.29a 3060 1 (𝜑 → ∃𝑥𝑃𝑦𝑃 𝐴 = (𝑥 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131   × cxp 5036   “ cima 5041  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  ≤Gcleg 25277 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by:  legso  25294
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