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Theorem ltgseg 26482
 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 784 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = 𝐴)
2 simpr 489 . . . . . 6 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
32fveq2d 6663 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = ( ‘⟨𝑥, 𝑦⟩))
41, 3eqtr3d 2796 . . . 4 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( ‘⟨𝑥, 𝑦⟩))
5 df-ov 7154 . . . 4 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
64, 5eqtr4di 2812 . . 3 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 𝑦))
7 simplr 769 . . . 4 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8 elxp2 5549 . . . 4 (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
97, 8sylib 221 . . 3 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
106, 9reximddv2 3203 . 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 2863 . . 3 (𝜑𝐴 ∈ ( “ (𝑃 × 𝑃)))
15 fvelima 6720 . . 3 ((Fun 𝐴 ∈ ( “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1611, 14, 15syl2anc 588 . 2 (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( 𝑎) = 𝐴)
1710, 16r19.29a 3214 1 (𝜑 → ∃𝑥𝑃𝑦𝑃 𝐴 = (𝑥 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∃wrex 3072  ⟨cop 4529   × cxp 5523   “ cima 5528  Fun wfun 6330  ‘cfv 6336  (class class class)co 7151  Basecbs 16534  distcds 16625  TarskiGcstrkg 26316  Itvcitv 26322  ≤Gcleg 26468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7154 This theorem is referenced by:  legso  26485
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