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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 31962 | . . 3 β’ (π β oMnd β π β Toset) | |
3 | tospos 18314 | . . 3 β’ (π β Toset β π β Poset) | |
4 | 1, 2, 3 | 3syl 18 | . 2 β’ (π β π β Poset) |
5 | omndmnd 31961 | . . . . 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 18569 | . . . 4 β’ ((π β Mnd β§ π β π΅ β§ π β π΅) β (π + π) β π΅) |
12 | 6, 7, 8, 11 | syl3anc 1372 | . . 3 β’ (π β (π + π) β π΅) |
13 | omndadd2d.w | . . . 4 β’ (π β π β π΅) | |
14 | 9, 10 | mndcl 18569 | . . . 4 β’ ((π β Mnd β§ π β π΅ β§ π β π΅) β (π + π) β π΅) |
15 | 6, 7, 13, 14 | syl3anc 1372 | . . 3 β’ (π β (π + π) β π΅) |
16 | omndadd2d.z | . . . 4 β’ (π β π β π΅) | |
17 | 9, 10 | mndcl 18569 | . . . 4 β’ ((π β Mnd β§ π β π΅ β§ π β π΅) β (π + π) β π΅) |
18 | 6, 16, 13, 17 | syl3anc 1372 | . . 3 β’ (π β (π + π) β π΅) |
19 | 12, 15, 18 | 3jca 1129 | . 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 31964 | . . 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 1384 | . 2 β’ (π β (π + π) β€ (π + π)) |
25 | omndadd2d.1 | . . 3 β’ (π β π β€ π) | |
26 | 9, 22, 10 | omndadd 31963 | . . 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 1384 | . 2 β’ (π β (π + π) β€ (π + π)) |
28 | 9, 22 | postr 18214 | . . 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 408 | . 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 838 | 1 β’ (π β (π + π) β€ (π + π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 lecple 17145 Posetcpo 18201 Tosetctos 18310 Mndcmnd 18561 oppgcoppg 19128 oMndcomnd 31954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-dec 12624 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-ple 17158 df-poset 18207 df-toset 18311 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-oppg 19129 df-omnd 31956 |
This theorem is referenced by: archiabllem2c 32080 |
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