Theorem ogrpaddlt 30259

Description: In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Hypotheses

Ref	Expression
ogrpaddlt.0	⊢ 𝐵 = (Base‘𝐺)
ogrpaddlt.1	⊢ < = (lt‘𝐺)
ogrpaddlt.2	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
ogrpaddlt	⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))

Proof of Theorem ogrpaddlt

Step	Hyp	Ref	Expression
1		isogrp 30243	. . . . 5 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 492	. . . 4 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	3ad2ant1 1167	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oMnd)
4		simp2 1171	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
5		simp1 1170	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oGrp)
6		simp21 1267	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 ∈ 𝐵)
7		simp22 1268	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑌 ∈ 𝐵)
8		simp3 1172	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
9		eqid 2825	. . . . . 6 ⊢ (le‘𝐺) = (le‘𝐺)
10		ogrpaddlt.1	. . . . . 6 ⊢ < = (lt‘𝐺)
11	9, 10	pltle 17321	. . . . 5 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → 𝑋(le‘𝐺)𝑌))
12	11	imp 397	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋(le‘𝐺)𝑌)
13	5, 6, 7, 8, 12	syl31anc 1496	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋(le‘𝐺)𝑌)
14		ogrpaddlt.0	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
15		ogrpaddlt.2	. . . 4 ⊢ + = (+g‘𝐺)
16	14, 9, 15	omndadd 30247	. . 3 ⊢ ((𝐺 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋(le‘𝐺)𝑌) → (𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍))
17	3, 4, 13, 16	syl3anc 1494	. 2 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍))
18	10	pltne 17322	. . . . 5 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → 𝑋 ≠ 𝑌))
19	18	imp 397	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 ≠ 𝑌)
20	5, 6, 7, 8, 19	syl31anc 1496	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 ≠ 𝑌)
21		ogrpgrp 30244	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
22	14, 15	grprcan 17816	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
23	22	biimpd 221	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
24	21, 23	sylan 575	. . . . 5 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
25	24	necon3d 3020	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≠ 𝑌 → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍)))
26	25	3impia 1149	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≠ 𝑌) → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))
27	5, 4, 20, 26	syl3anc 1494	. 2 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))
28		ovex 6942	. . . 4 ⊢ (𝑋 + 𝑍) ∈ V
29		ovex 6942	. . . 4 ⊢ (𝑌 + 𝑍) ∈ V
30	9, 10	pltval 17320	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 + 𝑍) ∈ V ∧ (𝑌 + 𝑍) ∈ V) → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
31	28, 29, 30	mp3an23 1581	. . 3 ⊢ (𝐺 ∈ oGrp → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
32	31	3ad2ant1 1167	. 2 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
33	17, 27, 32	mpbir2and 704	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  Vcvv 3414   class class class wbr 4875  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  +_gcplusg 16312  lecple 16319  ltcplt 17301  Grpcgrp 17783  oMndcomnd 30238  oGrpcogrp 30239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-riota 6871  df-ov 6913  df-0g 16462  df-plt 17318  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-omnd 30240  df-ogrp 30241

This theorem is referenced by:  ogrpaddltbi  30260  ogrpaddltrd  30261  ogrpinv0lt  30264  isarchi3  30282  archirngz  30284  archiabllem1b  30287  archiabllem2c  30290  ofldchr  30355