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Theorem legov3 28688

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 28687	. . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
13	12	orbi1d 923	. 2 ⊢ (𝜑 → (((𝐴 − 𝐵) < (𝐶 − 𝐷) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)) ↔ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))))
14		simprl 777	. . . 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 28677	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐴 − 𝐵))
16	15	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐴 − 𝐵))
17		simpr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
18	16, 17	breqtrd 5101	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
19	18	adantlr 722	. . . 4 ⊢ (((𝜑 ∧ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))) ∧ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
20		simpr 486	. . . 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 967	. . 3 ⊢ ((𝜑 ∧ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷))) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
22		simplr 775	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
23		simpr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷))
24	23	neqned 2943	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))
25	22, 24	jca 517	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) = (𝐶 − 𝐷)) → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)))
26	25	ex 414	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → (¬ (𝐴 − 𝐵) = (𝐶 − 𝐷) → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
27	26	orrd 870	. . . 4 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ∨ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))))
28	27	orcomd 878	. . 3 ⊢ ((𝜑 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)))
29	21, 28	impbida 807	. 2 ⊢ (𝜑 → ((((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) ∨ (𝐴 − 𝐵) = (𝐶 − 𝐷)) ↔ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)))
30	13, 29	bitr2d 282	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 208 ∧ wa 397 ∨ wo 854 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∖ cdif 3882 ⊆ wss 3885 class class class wbr 5075 I cid 5515 × cxp 5619 dom cdm 5621 ↾ cres 5623 “ cima 5624 Fun wfun 6483 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 distcds 17224 TarskiGcstrkg 28517 Itvcitv 28523 ≤Gcleg 28672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-xnn0 12506 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604 df-hash 14288 df-word 14471 df-concat 14528 df-s1 14554 df-s2 14805 df-s3 14806 df-trkgc 28538 df-trkgb 28539 df-trkgcb 28540 df-trkg 28543 df-cgrg 28601 df-leg 28673

This theorem is referenced by: legso 28689

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