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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 32728 | . . . . 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 18682 | . . . 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 12526 | . . . . . 6 β’ ((π β β0 β§ π β β0) β (π β€ π β (π β π) β β0)) | |
13 | 12 | biimpa 476 | . . . . 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 835 | . . . 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 19022 | . . 3 β’ (π β ((π β π) Β· π) β π΅) |
17 | 4, 8, 3, 9, 15 | mulgnn0cld 19022 | . . 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 32736 | . . . 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 1372 | . . 3 β’ (π β 0 β€ ((π β π) Β· π)) |
22 | eqid 2726 | . . . 4 β’ (+gβπ) = (+gβπ) | |
23 | 4, 19, 22 | omndadd 32730 | . . 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 1380 | . 2 β’ (π β ( 0 (+gβπ)(π Β· π)) β€ (((π β π) Β· π)(+gβπ)(π Β· π))) |
25 | 4, 22, 5 | mndlid 18687 | . . 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 583 | . 2 β’ (π β ( 0 (+gβπ)(π Β· π)) = (π Β· π)) |
27 | 4, 8, 22 | mulgnn0dir 19031 | . . . 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 1369 | . . 3 β’ (π β (((π β π) + π) Β· π) = (((π β π) Β· π)(+gβπ)(π Β· π))) |
29 | 10 | nn0cnd 12538 | . . . . 5 β’ (π β π β β) |
30 | 9 | nn0cnd 12538 | . . . . 5 β’ (π β π β β) |
31 | 29, 30 | npcand 11579 | . . . 4 β’ (π β ((π β π) + π) = π) |
32 | 31 | oveq1d 7420 | . . 3 β’ (π β (((π β π) + π) Β· π) = (π Β· π)) |
33 | 28, 32 | eqtr3d 2768 | . 2 β’ (π β (((π β π) Β· π)(+gβπ)(π Β· π)) = (π Β· π)) |
34 | 24, 26, 33 | 3brtr3d 5172 | 1 β’ (π β (π Β· π) β€ (π Β· π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 + caddc 11115 β€ cle 11253 β cmin 11448 β0cn0 12476 Basecbs 17153 +gcplusg 17206 lecple 17213 0gc0g 17394 Mndcmnd 18667 .gcmg 18995 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-rmo 3370 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-1st 7974 df-2nd 7975 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-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-seq 13973 df-0g 17396 df-proset 18260 df-poset 18278 df-toset 18382 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mulg 18996 df-omnd 32723 |
This theorem is referenced by: (None) |
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