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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 32729 | . . 3 β’ (π β oMnd β π β Toset) | |
| 3 | tospos 18385 | . . 3 β’ (π β Toset β π β Poset) | |
| 4 | 1, 2, 3 | 3syl 18 | . 2 β’ (π β π β Poset) |
| 5 | omndmnd 32728 | . . . . 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 18675 | . . . 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 18675 | . . . 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 18675 | . . . 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 32731 | . . 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 32730 | . . 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 18285 | . . 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 406 | . 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 836 | 1 β’ (π β (π + π) β€ (π + π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 lecple 17213 Posetcpo 18272 Tosetctos 18381 Mndcmnd 18667 oppgcoppg 19261 oMndcomnd 32721 |
| 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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-dec 12682 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-ple 17226 df-poset 18278 df-toset 18382 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-oppg 19262 df-omnd 32723 |
| This theorem is referenced by: archiabllem2c 32847 |
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