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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > omndadd2rd | Structured version Visualization version GIF version | ||
| Description: In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.) |
| Ref | Expression |
|---|---|
| omndadd.0 | β’ π΅ = (Baseβπ) |
| omndadd.1 | β’ β€ = (leβπ) |
| omndadd.2 | β’ + = (+gβπ) |
| omndadd2d.m | β’ (π β π β oMnd) |
| omndadd2d.w | β’ (π β π β π΅) |
| omndadd2d.x | β’ (π β π β π΅) |
| omndadd2d.y | β’ (π β π β π΅) |
| omndadd2d.z | β’ (π β π β π΅) |
| omndadd2d.1 | β’ (π β π β€ π) |
| omndadd2d.2 | β’ (π β π β€ π) |
| omndadd2rd.c | β’ (π β (oppgβπ) β oMnd) |
| Ref | Expression |
|---|---|
| omndadd2rd | β’ (π β (π + π) β€ (π + π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | omndadd2d.m | . . 3 β’ (π β π β oMnd) | |
| 2 | omndtos 32828 | . . 3 β’ (π β oMnd β π β Toset) | |
| 3 | tospos 18409 | . . 3 β’ (π β Toset β π β Poset) | |
| 4 | 1, 2, 3 | 3syl 18 | . 2 β’ (π β π β Poset) |
| 5 | omndmnd 32827 | . . . . 5 β’ (π β oMnd β π β Mnd) | |
| 6 | 1, 5 | syl 17 | . . . 4 β’ (π β π β Mnd) |
| 7 | omndadd2d.x | . . . 4 β’ (π β π β π΅) | |
| 8 | omndadd2d.y | . . . 4 β’ (π β π β π΅) | |
| 9 | omndadd.0 | . . . . 5 β’ π΅ = (Baseβπ) | |
| 10 | omndadd.2 | . . . . 5 β’ + = (+gβπ) | |
| 11 | 9, 10 | mndcl 18699 | . . . 4 β’ ((π β Mnd β§ π β π΅ β§ π β π΅) β (π + π) β π΅) |
| 12 | 6, 7, 8, 11 | syl3anc 1368 | . . 3 β’ (π β (π + π) β π΅) |
| 13 | omndadd2d.w | . . . 4 β’ (π β π β π΅) | |
| 14 | 9, 10 | mndcl 18699 | . . . 4 β’ ((π β Mnd β§ π β π΅ β§ π β π΅) β (π + π) β π΅) |
| 15 | 6, 7, 13, 14 | syl3anc 1368 | . . 3 β’ (π β (π + π) β π΅) |
| 16 | omndadd2d.z | . . . 4 β’ (π β π β π΅) | |
| 17 | 9, 10 | mndcl 18699 | . . . 4 β’ ((π β Mnd β§ π β π΅ β§ π β π΅) β (π + π) β π΅) |
| 18 | 6, 16, 13, 17 | syl3anc 1368 | . . 3 β’ (π β (π + π) β π΅) |
| 19 | 12, 15, 18 | 3jca 1125 | . 2 β’ (π β ((π + π) β π΅ β§ (π + π) β π΅ β§ (π + π) β π΅)) |
| 20 | omndadd2rd.c | . . 3 β’ (π β (oppgβπ) β oMnd) | |
| 21 | omndadd2d.2 | . . 3 β’ (π β π β€ π) | |
| 22 | omndadd.1 | . . . 4 β’ β€ = (leβπ) | |
| 23 | 9, 22, 10 | omndaddr 32830 | . . 3 β’ (((oppgβπ) β oMnd β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ π β€ π) β (π + π) β€ (π + π)) |
| 24 | 20, 8, 13, 7, 21, 23 | syl131anc 1380 | . 2 β’ (π β (π + π) β€ (π + π)) |
| 25 | omndadd2d.1 | . . 3 β’ (π β π β€ π) | |
| 26 | 9, 22, 10 | omndadd 32829 | . . 3 β’ ((π β oMnd β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ π β€ π) β (π + π) β€ (π + π)) |
| 27 | 1, 7, 16, 13, 25, 26 | syl131anc 1380 | . 2 β’ (π β (π + π) β€ (π + π)) |
| 28 | 9, 22 | postr 18309 | . . 3 β’ ((π β Poset β§ ((π + π) β π΅ β§ (π + π) β π΅ β§ (π + π) β π΅)) β (((π + π) β€ (π + π) β§ (π + π) β€ (π + π)) β (π + π) β€ (π + π))) |
| 29 | 28 | imp 405 | . 2 β’ (((π β Poset β§ ((π + π) β π΅ β§ (π + π) β π΅ β§ (π + π) β π΅)) β§ ((π + π) β€ (π + π) β§ (π + π) β€ (π + π))) β (π + π) β€ (π + π)) |
| 30 | 4, 19, 24, 27, 29 | syl22anc 837 | 1 β’ (π β (π + π) β€ (π + π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5143 βcfv 6542 (class class class)co 7415 Basecbs 17177 +gcplusg 17230 lecple 17237 Posetcpo 18296 Tosetctos 18405 Mndcmnd 18691 oppgcoppg 19298 oMndcomnd 32820 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-dec 12706 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-ple 17250 df-poset 18302 df-toset 18406 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-oppg 19299 df-omnd 32822 |
| This theorem is referenced by: archiabllem2c 32946 |
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