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Theorem ltgov 26505
 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 5043 . . . 4 ((𝐴 𝐵) < (𝐶 𝐷) ↔ (𝐴 𝐵)(( 𝐸) ∖ I )(𝐶 𝐷))
3 brdif 5090 . . . 4 ((𝐴 𝐵)(( 𝐸) ∖ I )(𝐶 𝐷) ↔ ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)))
42, 3bitri 278 . . 3 ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)))
5 ovex 7190 . . . . 5 (𝐶 𝐷) ∈ V
65brresi 5838 . . . 4 ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ↔ ((𝐴 𝐵) ∈ 𝐸 ∧ (𝐴 𝐵) (𝐶 𝐷)))
76anbi1i 626 . . 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 643 . . 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 300 . 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 7205 . . . . . 6 (𝜑 → (𝐴 𝐵) ∈ ( “ (𝑃 × 𝑃)))
15 legso.a . . . . . 6 𝐸 = ( “ (𝑃 × 𝑃))
1614, 15eleqtrrdi 2864 . . . . 5 (𝜑 → (𝐴 𝐵) ∈ 𝐸)
1716biantrurd 536 . . . 4 (𝜑 → (¬ (𝐴 𝐵) I (𝐶 𝐷) ↔ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))))
185ideq 5699 . . . . 5 ((𝐴 𝐵) I (𝐶 𝐷) ↔ (𝐴 𝐵) = (𝐶 𝐷))
1918necon3bbii 2999 . . . 4 (¬ (𝐴 𝐵) I (𝐶 𝐷) ↔ (𝐴 𝐵) ≠ (𝐶 𝐷))
2017, 19bitr3di 289 . . 3 (𝜑 → (((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)) ↔ (𝐴 𝐵) ≠ (𝐶 𝐷)))
2120anbi2d 631 . 2 (𝜑 → (((𝐴 𝐵) (𝐶 𝐷) ∧ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ (𝐴 𝐵) ≠ (𝐶 𝐷))))
229, 21syl5bb 286 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 209   ∧ wa 399   = wceq 1539   ∈ wcel 2112   ≠ wne 2952   ∖ cdif 3858   ⊆ wss 3861   class class class wbr 5037   I cid 5434   × cxp 5527  dom cdm 5529   ↾ cres 5531   “ cima 5532  Fun wfun 6335  ‘cfv 6341  (class class class)co 7157  Basecbs 16556  distcds 16647  TarskiGcstrkg 26338  Itvcitv 26344  ≤Gcleg 26490 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 5174  ax-nul 5181  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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-ne 2953  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-fv 6349  df-ov 7160 This theorem is referenced by:  legov3  26506  legso  26507
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