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Theorem ltgov 28650

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
legso.a	⊢ 𝐸 = ( − “ (𝑃 × 𝑃))
legso.f	⊢ (𝜑 → Fun − )
legso.l	⊢ < = (( ≤ ↾ 𝐸) ∖ I )
legso.d	⊢ (𝜑 → (𝑃 × 𝑃) ⊆ dom − )
ltgov.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
ltgov.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

**Assertion**
Ref	Expression
ltgov	⊢ (𝜑 → ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))

**Proof of Theorem ltgov**
Step	Hyp	Ref	Expression
1		legso.l	. . . . 5 ⊢ < = (( ≤ ↾ 𝐸) ∖ I )
2	1	breqi 5103	. . . 4 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ (𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷))
3		brdif 5150	. . . 4 ⊢ ((𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))
4	2, 3	bitri 275	. . 3 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))
5		ovex 7391	. . . . 5 ⊢ (𝐶 − 𝐷) ∈ V
6	5	brresi 5946	. . . 4 ⊢ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)))
7	6	anbi1i 625	. . 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 645	. . 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))))
9	4, 7, 8	3bitri 297	. 2 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))))
10		ltgov.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11		ltgov.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12		legso.f	. . . . . . 7 ⊢ (𝜑 → Fun − )
13		legso.d	. . . . . . 7 ⊢ (𝜑 → (𝑃 × 𝑃) ⊆ dom − )
14	10, 11, 12, 13	elovimad 7408	. . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ( − “ (𝑃 × 𝑃)))
15		legso.a	. . . . . 6 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃))
16	14, 15	eleqtrrdi 2846	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝐸)
17	16	biantrurd 532	. . . 4 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))))
18	5	ideq 5800	. . . . 5 ⊢ ((𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷))
19	18	necon3bbii 2978	. . . 4 ⊢ (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))
20	17, 19	bitr3di 286	. . 3 ⊢ (𝜑 → (((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)))
21	20	anbi2d 631	. 2 ⊢ (𝜑 → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
22	9, 21	bitrid 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 1542 ∈ wcel 2114 ≠ wne 2931 ∖ cdif 3897 ⊆ wss 3900 class class class wbr 5097 I cid 5517 × cxp 5621 dom cdm 5623 ↾ cres 5625 “ cima 5626 Fun wfun 6485 ‘cfv 6491 (class class class)co 7358 Basecbs 17138 distcds 17188 TarskiGcstrkg 28480 Itvcitv 28486 ≤Gcleg 28635

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-12 2183 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pr 5376

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3399 df-v 3441 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4285 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6447 df-fun 6493 df-fn 6494 df-fv 6499 df-ov 7361

This theorem is referenced by: legov3 28651 legso 28652

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