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Theorem legov3 28577

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 28576	. . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
13	12	orbi1d 916	. 2 ⊢ (𝜑 → (((𝐴 − 𝐵) < (𝐶 − 𝐷) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)) ↔ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))))
14		simprl 770	. . . 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 28566	. . . . . . 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 5145	. . . . 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 960	. . 3 ⊢ ((𝜑 ∧ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
22		simplr 768	. . . . . . 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 2939	. . . . . . 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 863	. . . 4 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ∨ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
28	27	orcomd 871	. . 3 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)))
29	21, 28	impbida 800	. 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 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∖ cdif 3923 ⊆ wss 3926 class class class wbr 5119 I cid 5547 × cxp 5652 dom cdm 5654 ↾ cres 5656 “ cima 5657 Fun wfun 6525 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 distcds 17280 TarskiGcstrkg 28406 Itvcitv 28412 ≤Gcleg 28561

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8719 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-concat 14589 df-s1 14614 df-s2 14867 df-s3 14868 df-trkgc 28427 df-trkgb 28428 df-trkgcb 28429 df-trkg 28432 df-cgrg 28490 df-leg 28562

This theorem is referenced by: legso 28578

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