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Theorem ltgov 28606
Description: Strict "shorter than" geometric relation between segments. (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 )
legso.l < = (( 𝐸) ∖ I )
legso.d (𝜑 → (𝑃 × 𝑃) ⊆ dom )
ltgov.a (𝜑𝐴𝑃)
ltgov.b (𝜑𝐵𝑃)
Assertion
Ref Expression
ltgov (𝜑 → ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ (𝐴 𝐵) ≠ (𝐶 𝐷))))

Proof of Theorem ltgov
StepHypRef Expression
1 legso.l . . . . 5 < = (( 𝐸) ∖ I )
21breqi 5148 . . . 4 ((𝐴 𝐵) < (𝐶 𝐷) ↔ (𝐴 𝐵)(( 𝐸) ∖ I )(𝐶 𝐷))
3 brdif 5195 . . . 4 ((𝐴 𝐵)(( 𝐸) ∖ I )(𝐶 𝐷) ↔ ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)))
42, 3bitri 275 . . 3 ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)))
5 ovex 7465 . . . . 5 (𝐶 𝐷) ∈ V
65brresi 6005 . . . 4 ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ↔ ((𝐴 𝐵) ∈ 𝐸 ∧ (𝐴 𝐵) (𝐶 𝐷)))
76anbi1i 624 . . 3 (((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)) ↔ (((𝐴 𝐵) ∈ 𝐸 ∧ (𝐴 𝐵) (𝐶 𝐷)) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)))
8 an21 644 . . 3 ((((𝐴 𝐵) ∈ 𝐸 ∧ (𝐴 𝐵) (𝐶 𝐷)) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))))
94, 7, 83bitri 297 . 2 ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))))
10 ltgov.a . . . . . . 7 (𝜑𝐴𝑃)
11 ltgov.b . . . . . . 7 (𝜑𝐵𝑃)
12 legso.f . . . . . . 7 (𝜑 → Fun )
13 legso.d . . . . . . 7 (𝜑 → (𝑃 × 𝑃) ⊆ dom )
1410, 11, 12, 13elovimad 7482 . . . . . 6 (𝜑 → (𝐴 𝐵) ∈ ( “ (𝑃 × 𝑃)))
15 legso.a . . . . . 6 𝐸 = ( “ (𝑃 × 𝑃))
1614, 15eleqtrrdi 2851 . . . . 5 (𝜑 → (𝐴 𝐵) ∈ 𝐸)
1716biantrurd 532 . . . 4 (𝜑 → (¬ (𝐴 𝐵) I (𝐶 𝐷) ↔ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))))
185ideq 5862 . . . . 5 ((𝐴 𝐵) I (𝐶 𝐷) ↔ (𝐴 𝐵) = (𝐶 𝐷))
1918necon3bbii 2987 . . . 4 (¬ (𝐴 𝐵) I (𝐶 𝐷) ↔ (𝐴 𝐵) ≠ (𝐶 𝐷))
2017, 19bitr3di 286 . . 3 (𝜑 → (((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)) ↔ (𝐴 𝐵) ≠ (𝐶 𝐷)))
2120anbi2d 630 . 2 (𝜑 → (((𝐴 𝐵) (𝐶 𝐷) ∧ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ (𝐴 𝐵) ≠ (𝐶 𝐷))))
229, 21bitrid 283 1 (𝜑 → ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ (𝐴 𝐵) ≠ (𝐶 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  cdif 3947  wss 3950   class class class wbr 5142   I cid 5576   × cxp 5682  dom cdm 5684  cres 5686  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-ne 2940  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-fn 6563  df-fv 6568  df-ov 7435
This theorem is referenced by:  legov3  28607  legso  28608
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