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Theorem ltgov 25712

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 4792	. . . 4 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ (𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷))
3		brdif 4839	. . . 4 ⊢ ((𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))
4	2, 3	bitri 264	. . 3 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))
5		ovex 6822	. . . . 5 ⊢ (𝐶 − 𝐷) ∈ V
6	5	brres 5543	. . . 4 ⊢ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ∈ 𝐸))
7	6	anbi1i 602	. . 3 ⊢ (((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ∈ 𝐸) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))
8		anass 459	. . 3 ⊢ ((((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ∈ 𝐸) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))))
9	4, 7, 8	3bitri 286	. 2 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))))
10	5	ideq 5413	. . . . 5 ⊢ ((𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷))
11	10	necon3bbii 2990	. . . 4 ⊢ (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))
12		ltgov.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
13		ltgov.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
14		legso.f	. . . . . . 7 ⊢ (𝜑 → Fun − )
15		legso.d	. . . . . . 7 ⊢ (𝜑 → (𝑃 × 𝑃) ⊆ dom − )
16	12, 13, 14, 15	elovimad 6837	. . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ( − “ (𝑃 × 𝑃)))
17		legso.a	. . . . . 6 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃))
18	16, 17	syl6eleqr 2861	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝐸)
19	18	biantrurd 516	. . . 4 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))))
20	11, 19	syl5rbbr 275	. . 3 ⊢ (𝜑 → (((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)))
21	20	anbi2d 606	. 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 272	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 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 ⊆ wss 3723 class class class wbr 4786 I cid 5156 × cxp 5247 dom cdm 5249 ↾ cres 5251 “ cima 5252 Fun wfun 6025 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 distcds 16157 TarskiGcstrkg 25549 Itvcitv 25555 ≤Gcleg 25697

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-fv 6039 df-ov 6795

This theorem is referenced by: legov3 25713 legso 25714

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