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Theorem ltgseg 25386
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 806 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = 𝐴)
2 simpr 477 . . . . . 6 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
32fveq2d 6154 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = ( ‘⟨𝑥, 𝑦⟩))
41, 3eqtr3d 2662 . . . 4 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( ‘⟨𝑥, 𝑦⟩))
5 df-ov 6608 . . . 4 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
64, 5syl6eqr 2678 . . 3 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 𝑦))
7 simplr 791 . . . 4 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8 elxp2 5097 . . . 4 (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
97, 8sylib 208 . . 3 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
106, 9reximddv2 3018 . 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 2714 . . 3 (𝜑𝐴 ∈ ( “ (𝑃 × 𝑃)))
15 fvelima 6206 . . 3 ((Fun 𝐴 ∈ ( “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1611, 14, 15syl2anc 692 . 2 (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1710, 16r19.29a 3076 1 (𝜑 → ∃𝑥𝑃𝑦𝑃 𝐴 = (𝑥 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wrex 2913  cop 4159   × cxp 5077  cima 5082  Fun wfun 5844  cfv 5850  (class class class)co 6605  Basecbs 15776  distcds 15866  TarskiGcstrkg 25224  Itvcitv 25230  ≤Gcleg 25372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608
This theorem is referenced by:  legso  25389
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