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Theorem legov3 28532

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 28531	. . 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 28521	. . . . . . 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 5136	. . . . 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 2933	. . . . . . 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 2109 ≠ wne 2926 ∖ cdif 3914 ⊆ wss 3917 class class class wbr 5110 I cid 5535 × cxp 5639 dom cdm 5641 ↾ cres 5643 “ cima 5644 Fun wfun 6508 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 distcds 17236 TarskiGcstrkg 28361 Itvcitv 28367 ≤Gcleg 28516

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8674 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-s2 14821 df-s3 14822 df-trkgc 28382 df-trkgb 28383 df-trkgcb 28384 df-trkg 28387 df-cgrg 28445 df-leg 28517

This theorem is referenced by: legso 28533

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