Theorem omndmul3 29713

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.)

Hypotheses

Ref	Expression
omndmul.0	|- B = ( Base ` M )
omndmul.1	|- .<_ = ( le ` M )
omndmul3.m	|- .x. = (.g ` M )
omndmul3.0	|- .0. = ( 0g ` M )
omndmul3.o	|- ( ph -> M e. oMnd )
omndmul3.1	|- ( ph -> N e. NN0 )
omndmul3.2	|- ( ph -> P e. NN0 )
omndmul3.3	|- ( ph -> N <_ P )
omndmul3.4	|- ( ph -> X e. B )
omndmul3.5	|- ( ph -> .0. .<_ X )

Assertion

Ref	Expression
omndmul3	|- ( ph -> ( N .x. X ) .<_ ( P .x. X ) )

Proof of Theorem omndmul3

Step	Hyp	Ref	Expression
1		omndmul3.o	. . 3 |- ( ph -> M e. oMnd )
2		omndmnd 29704	. . . . 5 |- ( M e. oMnd -> M e. Mnd )
3	1, 2	syl 17	. . . 4 |- ( ph -> M e. Mnd )
4		omndmul.0	. . . . 5 |- B = ( Base ` M )
5		omndmul3.0	. . . . 5 |- .0. = ( 0g ` M )
6	4, 5	mndidcl 17308	. . . 4 |- ( M e. Mnd -> .0. e. B )
7	3, 6	syl 17	. . 3 |- ( ph -> .0. e. B )
8		omndmul3.1	. . . . 5 |- ( ph -> N e. NN0 )
9		omndmul3.2	. . . . 5 |- ( ph -> P e. NN0 )
10		omndmul3.3	. . . . 5 |- ( ph -> N <_ P )
11		nn0sub 11343	. . . . . 6 |- ( ( N e. NN0 /\ P e. NN0 ) -> ( N <_ P <-> ( P - N ) e. NN0 ) )
12	11	biimpa 501	. . . . 5 |- ( ( ( N e. NN0 /\ P e. NN0 ) /\ N <_ P ) -> ( P - N ) e. NN0 )
13	8, 9, 10, 12	syl21anc 1325	. . . 4 |- ( ph -> ( P - N ) e. NN0 )
14		omndmul3.4	. . . 4 |- ( ph -> X e. B )
15		omndmul3.m	. . . . 5 |- .x. = (.g ` M )
16	4, 15	mulgnn0cl 17558	. . . 4 |- ( ( M e. Mnd /\ ( P - N ) e. NN0 /\ X e. B ) -> ( ( P - N ) .x. X ) e. B )
17	3, 13, 14, 16	syl3anc 1326	. . 3 |- ( ph -> ( ( P - N ) .x. X ) e. B )
18	4, 15	mulgnn0cl 17558	. . . 4 |- ( ( M e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B )
19	3, 8, 14, 18	syl3anc 1326	. . 3 |- ( ph -> ( N .x. X ) e. B )
20		omndmul3.5	. . . 4 |- ( ph -> .0. .<_ X )
21		omndmul.1	. . . . 5 |- .<_ = ( le ` M )
22	4, 21, 15, 5	omndmul2 29712	. . . 4 |- ( ( M e. oMnd /\ ( X e. B /\ ( P - N ) e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( ( P - N ) .x. X ) )
23	1, 14, 13, 20, 22	syl121anc 1331	. . 3 |- ( ph -> .0. .<_ ( ( P - N ) .x. X ) )
24		eqid 2622	. . . 4 |- ( +g ` M ) = ( +g ` M )
25	4, 21, 24	omndadd 29706	. . 3 |- ( ( M e. oMnd /\ ( .0. e. B /\ ( ( P - N ) .x. X ) e. B /\ ( N .x. X ) e. B ) /\ .0. .<_ ( ( P - N ) .x. X ) ) -> ( .0. ( +g ` M ) ( N .x. X ) ) .<_ ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) )
26	1, 7, 17, 19, 23, 25	syl131anc 1339	. 2 |- ( ph -> ( .0. ( +g ` M ) ( N .x. X ) ) .<_ ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) )
27	4, 24, 5	mndlid 17311	. . 3 |- ( ( M e. Mnd /\ ( N .x. X ) e. B ) -> ( .0. ( +g ` M ) ( N .x. X ) ) = ( N .x. X ) )
28	3, 19, 27	syl2anc 693	. 2 |- ( ph -> ( .0. ( +g ` M ) ( N .x. X ) ) = ( N .x. X ) )
29	4, 15, 24	mulgnn0dir 17571	. . . 4 |- ( ( M e. Mnd /\ ( ( P - N ) e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( ( P - N ) + N ) .x. X ) = ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) )
30	3, 13, 8, 14, 29	syl13anc 1328	. . 3 |- ( ph -> ( ( ( P - N ) + N ) .x. X ) = ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) )
31	9	nn0cnd 11353	. . . . 5 |- ( ph -> P e. CC )
32	8	nn0cnd 11353	. . . . 5 |- ( ph -> N e. CC )
33	31, 32	npcand 10396	. . . 4 |- ( ph -> ( ( P - N ) + N ) = P )
34	33	oveq1d 6665	. . 3 |- ( ph -> ( ( ( P - N ) + N ) .x. X ) = ( P .x. X ) )
35	30, 34	eqtr3d 2658	. 2 |- ( ph -> ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) = ( P .x. X ) )
36	26, 28, 35	3brtr3d 4684	1 |- ( ph -> ( N .x. X ) .<_ ( P .x. X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   + caddc 9939   <_ cle 10075   - cmin 10266   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   lecple 15948   0gc0g 16100   Mndcmnd 17294  ._gcmg 17540  oMndcomnd 29697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-0g 16102  df-preset 16928  df-poset 16946  df-toset 17034  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mulg 17541  df-omnd 29699

This theorem is referenced by: (None)