MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltgseg Structured version   Visualization version   GIF version

Theorem ltgseg 28605
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 783 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = 𝐴)
2 simpr 484 . . . . . 6 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
32fveq2d 6909 . . . . 5 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( 𝑎) = ( ‘⟨𝑥, 𝑦⟩))
41, 3eqtr3d 2778 . . . 4 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( ‘⟨𝑥, 𝑦⟩))
5 df-ov 7435 . . . 4 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
64, 5eqtr4di 2794 . . 3 ((((((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 𝑦))
7 simplr 768 . . . 4 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8 elxp2 5708 . . . 4 (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
97, 8sylib 218 . . 3 (((𝜑𝑎 ∈ (𝑃 × 𝑃)) ∧ ( 𝑎) = 𝐴) → ∃𝑥𝑃𝑦𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
106, 9reximddv2 3214 . 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 2850 . . 3 (𝜑𝐴 ∈ ( “ (𝑃 × 𝑃)))
15 fvelima 6973 . . 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 395   = wceq 1539  wcel 2107  wrex 3069  cop 4631   × cxp 5682  cima 5687  Fun wfun 6554  cfv 6560  (class class class)co 7432  Basecbs 17248  distcds 17307  TarskiGcstrkg 28436  Itvcitv 28442  ≤Gcleg 28591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435
This theorem is referenced by:  legso  28608
  Copyright terms: Public domain W3C validator