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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	⊢ (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 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)))
4	2, 3	bitri 278	. . 3 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))
5		ovex 7190	. . . . 5 ⊢ (𝐶 − 𝐷) ∈ V
6	5	brresi 5838	. . . 4 ⊢ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)))
7	6	anbi1i 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))))
9	4, 7, 8	3bitri 300	. 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 7205	. . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ( − “ (𝑃 × 𝑃)))
15		legso.a	. . . . . 6 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃))
16	14, 15	eleqtrrdi 2864	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝐸)
17	16	biantrurd 536	. . . 4 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))))
18	5	ideq 5699	. . . . 5 ⊢ ((𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷))
19	18	necon3bbii 2999	. . . 4 ⊢ (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))
20	17, 19	bitr3di 289	. . 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	syl5bb 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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