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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omndmul3 | Structured version Visualization version GIF version |
Description: In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
omndmul.0 | β’ π΅ = (Baseβπ) |
omndmul.1 | β’ β€ = (leβπ) |
omndmul3.m | β’ Β· = (.gβπ) |
omndmul3.0 | β’ 0 = (0gβπ) |
omndmul3.o | β’ (π β π β oMnd) |
omndmul3.1 | β’ (π β π β β0) |
omndmul3.2 | β’ (π β π β β0) |
omndmul3.3 | β’ (π β π β€ π) |
omndmul3.4 | β’ (π β π β π΅) |
omndmul3.5 | β’ (π β 0 β€ π) |
Ref | Expression |
---|---|
omndmul3 | β’ (π β (π Β· π) β€ (π Β· π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omndmul3.o | . . 3 β’ (π β π β oMnd) | |
2 | omndmnd 31961 | . . . . 5 β’ (π β oMnd β π β Mnd) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β π β Mnd) |
4 | omndmul.0 | . . . . 5 β’ π΅ = (Baseβπ) | |
5 | omndmul3.0 | . . . . 5 β’ 0 = (0gβπ) | |
6 | 4, 5 | mndidcl 18576 | . . . 4 β’ (π β Mnd β 0 β π΅) |
7 | 3, 6 | syl 17 | . . 3 β’ (π β 0 β π΅) |
8 | omndmul3.m | . . . 4 β’ Β· = (.gβπ) | |
9 | omndmul3.1 | . . . . 5 β’ (π β π β β0) | |
10 | omndmul3.2 | . . . . 5 β’ (π β π β β0) | |
11 | omndmul3.3 | . . . . 5 β’ (π β π β€ π) | |
12 | nn0sub 12468 | . . . . . 6 β’ ((π β β0 β§ π β β0) β (π β€ π β (π β π) β β0)) | |
13 | 12 | biimpa 478 | . . . . 5 β’ (((π β β0 β§ π β β0) β§ π β€ π) β (π β π) β β0) |
14 | 9, 10, 11, 13 | syl21anc 837 | . . . 4 β’ (π β (π β π) β β0) |
15 | omndmul3.4 | . . . 4 β’ (π β π β π΅) | |
16 | 4, 8, 3, 14, 15 | mulgnn0cld 18902 | . . 3 β’ (π β ((π β π) Β· π) β π΅) |
17 | 4, 8, 3, 9, 15 | mulgnn0cld 18902 | . . 3 β’ (π β (π Β· π) β π΅) |
18 | omndmul3.5 | . . . 4 β’ (π β 0 β€ π) | |
19 | omndmul.1 | . . . . 5 β’ β€ = (leβπ) | |
20 | 4, 19, 8, 5 | omndmul2 31969 | . . . 4 β’ ((π β oMnd β§ (π β π΅ β§ (π β π) β β0) β§ 0 β€ π) β 0 β€ ((π β π) Β· π)) |
21 | 1, 15, 14, 18, 20 | syl121anc 1376 | . . 3 β’ (π β 0 β€ ((π β π) Β· π)) |
22 | eqid 2733 | . . . 4 β’ (+gβπ) = (+gβπ) | |
23 | 4, 19, 22 | omndadd 31963 | . . 3 β’ ((π β oMnd β§ ( 0 β π΅ β§ ((π β π) Β· π) β π΅ β§ (π Β· π) β π΅) β§ 0 β€ ((π β π) Β· π)) β ( 0 (+gβπ)(π Β· π)) β€ (((π β π) Β· π)(+gβπ)(π Β· π))) |
24 | 1, 7, 16, 17, 21, 23 | syl131anc 1384 | . 2 β’ (π β ( 0 (+gβπ)(π Β· π)) β€ (((π β π) Β· π)(+gβπ)(π Β· π))) |
25 | 4, 22, 5 | mndlid 18581 | . . 3 β’ ((π β Mnd β§ (π Β· π) β π΅) β ( 0 (+gβπ)(π Β· π)) = (π Β· π)) |
26 | 3, 17, 25 | syl2anc 585 | . 2 β’ (π β ( 0 (+gβπ)(π Β· π)) = (π Β· π)) |
27 | 4, 8, 22 | mulgnn0dir 18911 | . . . 4 β’ ((π β Mnd β§ ((π β π) β β0 β§ π β β0 β§ π β π΅)) β (((π β π) + π) Β· π) = (((π β π) Β· π)(+gβπ)(π Β· π))) |
28 | 3, 14, 9, 15, 27 | syl13anc 1373 | . . 3 β’ (π β (((π β π) + π) Β· π) = (((π β π) Β· π)(+gβπ)(π Β· π))) |
29 | 10 | nn0cnd 12480 | . . . . 5 β’ (π β π β β) |
30 | 9 | nn0cnd 12480 | . . . . 5 β’ (π β π β β) |
31 | 29, 30 | npcand 11521 | . . . 4 β’ (π β ((π β π) + π) = π) |
32 | 31 | oveq1d 7373 | . . 3 β’ (π β (((π β π) + π) Β· π) = (π Β· π)) |
33 | 28, 32 | eqtr3d 2775 | . 2 β’ (π β (((π β π) Β· π)(+gβπ)(π Β· π)) = (π Β· π)) |
34 | 24, 26, 33 | 3brtr3d 5137 | 1 β’ (π β (π Β· π) β€ (π Β· π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 + caddc 11059 β€ cle 11195 β cmin 11390 β0cn0 12418 Basecbs 17088 +gcplusg 17138 lecple 17145 0gc0g 17326 Mndcmnd 18561 .gcmg 18877 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-rmo 3352 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-1st 7922 df-2nd 7923 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-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-seq 13913 df-0g 17328 df-proset 18189 df-poset 18207 df-toset 18311 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mulg 18878 df-omnd 31956 |
This theorem is referenced by: (None) |
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