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Theorem ltgov 26101
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 4932 . . . 4 ((𝐴 𝐵) < (𝐶 𝐷) ↔ (𝐴 𝐵)(( 𝐸) ∖ I )(𝐶 𝐷))
3 brdif 4979 . . . 4 ((𝐴 𝐵)(( 𝐸) ∖ I )(𝐶 𝐷) ↔ ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)))
42, 3bitri 267 . . 3 ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)))
5 ovex 7007 . . . . 5 (𝐶 𝐷) ∈ V
65brresi 5702 . . . 4 ((𝐴 𝐵)( 𝐸)(𝐶 𝐷) ↔ ((𝐴 𝐵) ∈ 𝐸 ∧ (𝐴 𝐵) (𝐶 𝐷)))
76anbi1i 615 . . 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 632 . . 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 289 . 2 ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))))
105ideq 5570 . . . . 5 ((𝐴 𝐵) I (𝐶 𝐷) ↔ (𝐴 𝐵) = (𝐶 𝐷))
1110necon3bbii 3009 . . . 4 (¬ (𝐴 𝐵) I (𝐶 𝐷) ↔ (𝐴 𝐵) ≠ (𝐶 𝐷))
12 ltgov.a . . . . . . 7 (𝜑𝐴𝑃)
13 ltgov.b . . . . . . 7 (𝜑𝐵𝑃)
14 legso.f . . . . . . 7 (𝜑 → Fun )
15 legso.d . . . . . . 7 (𝜑 → (𝑃 × 𝑃) ⊆ dom )
1612, 13, 14, 15elovimad 7022 . . . . . 6 (𝜑 → (𝐴 𝐵) ∈ ( “ (𝑃 × 𝑃)))
17 legso.a . . . . . 6 𝐸 = ( “ (𝑃 × 𝑃))
1816, 17syl6eleqr 2872 . . . . 5 (𝜑 → (𝐴 𝐵) ∈ 𝐸)
1918biantrurd 525 . . . 4 (𝜑 → (¬ (𝐴 𝐵) I (𝐶 𝐷) ↔ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))))
2011, 19syl5rbbr 278 . . 3 (𝜑 → (((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷)) ↔ (𝐴 𝐵) ≠ (𝐶 𝐷)))
2120anbi2d 620 . 2 (𝜑 → (((𝐴 𝐵) (𝐶 𝐷) ∧ ((𝐴 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 𝐵) I (𝐶 𝐷))) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ (𝐴 𝐵) ≠ (𝐶 𝐷))))
229, 21syl5bb 275 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 198  wa 387   = wceq 1508  wcel 2051  wne 2962  cdif 3821  wss 3824   class class class wbr 4926   I cid 5308   × cxp 5402  dom cdm 5404  cres 5406  cima 5407  Fun wfun 6180  cfv 6186  (class class class)co 6975  Basecbs 16338  distcds 16429  TarskiGcstrkg 25934  Itvcitv 25940  ≤Gcleg 26086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-fv 6194  df-ov 6978
This theorem is referenced by:  legov3  26102  legso  26103
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