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Theorem legov3 28606

Description: An equivalent definition of the less-than relationship, from the strict relation. (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
legov3	⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))))

**Proof of Theorem legov3**
Step	Hyp	Ref	Expression
1		legval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		legval.d	. . . 4 ⊢ − = (dist‘𝐺)
3		legval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		legval.l	. . . 4 ⊢ ≤ = (≤G‘𝐺)
5		legval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		legso.a	. . . 4 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃))
7		legso.f	. . . 4 ⊢ (𝜑 → Fun − )
8		legso.l	. . . 4 ⊢ < = (( ≤ ↾ 𝐸) ∖ I )
9		legso.d	. . . 4 ⊢ (𝜑 → (𝑃 × 𝑃) ⊆ dom − )
10		ltgov.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11		ltgov.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	ltgov 28605	. . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
13	12	orbi1d 917	. 2 ⊢ (𝜑 → (((𝐴 − 𝐵) < (𝐶 − 𝐷) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)) ↔ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))))
14		simprl 771	. . . 4 ⊢ (((𝜑 ∧ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))) ∧ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
15	1, 2, 3, 4, 5, 10, 11	legid 28595	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐴 − 𝐵))
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐴 − 𝐵))
17		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
18	16, 17	breqtrd 5169	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
19	18	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))) ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
20		simpr 484	. . . 4 ⊢ ((𝜑 ∧ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))) → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)))
21	14, 19, 20	mpjaodan 961	. . 3 ⊢ ((𝜑 ∧ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
22		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
23		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷))
24	23	neqned 2947	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))
25	22, 24	jca 511	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)))
26	25	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → (¬ (𝐴 − 𝐵) = (𝐶 − 𝐷) → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
27	26	orrd 864	. . . 4 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ∨ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
28	27	orcomd 872	. . 3 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)))
29	21, 28	impbida 801	. 2 ⊢ (𝜑 → ((((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)) ↔ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)))
30	13, 29	bitr2d 280	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 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 I cid 5577 × cxp 5683 dom cdm 5685 ↾ cres 5687 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 ≤Gcleg 28590

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-cgrg 28519 df-leg 28591

This theorem is referenced by: legso 28607

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