Theorem omndmul2 29712

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 )
omndmul2.2	|- .x. = (.g ` M )
omndmul2.3	|- .0. = ( 0g ` M )

Assertion

Ref	Expression
omndmul2	|- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )

Proof of Theorem omndmul2

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 1039	. . 3 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) /\ .0. .<_ X ) <-> ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) /\ .0. .<_ X ) )
2		anass 681	. . . 4 |- ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) <-> ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) )
3	2	anbi1i 731	. . 3 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) <-> ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) /\ .0. .<_ X ) )
4	1, 3	bitr4i 267	. 2 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) /\ .0. .<_ X ) <-> ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) )
5		simplr 792	. . 3 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> N e. NN0 )
6		oveq1 6657	. . . . 5 |- ( m = 0 -> ( m .x. X ) = ( 0 .x. X ) )
7	6	breq2d 4665	. . . 4 |- ( m = 0 -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( 0 .x. X ) ) )
8		oveq1 6657	. . . . 5 |- ( m = n -> ( m .x. X ) = ( n .x. X ) )
9	8	breq2d 4665	. . . 4 |- ( m = n -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( n .x. X ) ) )
10		oveq1 6657	. . . . 5 |- ( m = ( n + 1 ) -> ( m .x. X ) = ( ( n + 1 ) .x. X ) )
11	10	breq2d 4665	. . . 4 |- ( m = ( n + 1 ) -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( ( n + 1 ) .x. X ) ) )
12		oveq1 6657	. . . . 5 |- ( m = N -> ( m .x. X ) = ( N .x. X ) )
13	12	breq2d 4665	. . . 4 |- ( m = N -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( N .x. X ) ) )
14		omndtos 29705	. . . . . . . 8 |- ( M e. oMnd -> M e. Toset )
15		tospos 29658	. . . . . . . 8 |- ( M e. Toset -> M e. Poset )
16	14, 15	syl 17	. . . . . . 7 |- ( M e. oMnd -> M e. Poset )
17		omndmnd 29704	. . . . . . . 8 |- ( M e. oMnd -> M e. Mnd )
18		omndmul.0	. . . . . . . . 9 |- B = ( Base ` M )
19		omndmul2.3	. . . . . . . . 9 |- .0. = ( 0g ` M )
20	18, 19	mndidcl 17308	. . . . . . . 8 |- ( M e. Mnd -> .0. e. B )
21	17, 20	syl 17	. . . . . . 7 |- ( M e. oMnd -> .0. e. B )
22		omndmul.1	. . . . . . . 8 |- .<_ = ( le ` M )
23	18, 22	posref 16951	. . . . . . 7 |- ( ( M e. Poset /\ .0. e. B ) -> .0. .<_ .0. )
24	16, 21, 23	syl2anc 693	. . . . . 6 |- ( M e. oMnd -> .0. .<_ .0. )
25	24	ad3antrrr 766	. . . . 5 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ .0. )
26		omndmul2.2	. . . . . . 7 |- .x. = (.g ` M )
27	18, 19, 26	mulg0 17546	. . . . . 6 |- ( X e. B -> ( 0 .x. X ) = .0. )
28	27	ad3antlr 767	. . . . 5 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> ( 0 .x. X ) = .0. )
29	25, 28	breqtrrd 4681	. . . 4 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( 0 .x. X ) )
30	16	ad5antr 770	. . . . 5 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> M e. Poset )
31	17	ad5antr 770	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> M e. Mnd )
32	31, 20	syl 17	. . . . . 6 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> .0. e. B )
33		simplr 792	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> n e. NN0 )
34		simp-5r 809	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> X e. B )
35	18, 26	mulgnn0cl 17558	. . . . . . 7 |- ( ( M e. Mnd /\ n e. NN0 /\ X e. B ) -> ( n .x. X ) e. B )
36	31, 33, 34, 35	syl3anc 1326	. . . . . 6 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( n .x. X ) e. B )
37		simpr32 1152	. . . . . . . . . 10 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> n e. NN0 )
38		1nn0 11308	. . . . . . . . . . 11 |- 1 e. NN0
39	38	a1i 11	. . . . . . . . . 10 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> 1 e. NN0 )
40	37, 39	nn0addcld 11355	. . . . . . . . 9 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> ( n + 1 ) e. NN0 )
41	40	3anassrs 1290	. . . . . . . 8 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) -> ( n + 1 ) e. NN0 )
42	41	3anassrs 1290	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( n + 1 ) e. NN0 )
43	18, 26	mulgnn0cl 17558	. . . . . . 7 |- ( ( M e. Mnd /\ ( n + 1 ) e. NN0 /\ X e. B ) -> ( ( n + 1 ) .x. X ) e. B )
44	31, 42, 34, 43	syl3anc 1326	. . . . . 6 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( ( n + 1 ) .x. X ) e. B )
45	32, 36, 44	3jca 1242	. . . . 5 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( .0. e. B /\ ( n .x. X ) e. B /\ ( ( n + 1 ) .x. X ) e. B ) )
46		simpr 477	. . . . 5 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> .0. .<_ ( n .x. X ) )
47		simp-4l 806	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. oMnd )
48	17	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. Mnd )
49	48, 20	syl 17	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. e. B )
50		simp-4r 807	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> X e. B )
51		simpr 477	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> n e. NN0 )
52	48, 51, 50, 35	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( n .x. X ) e. B )
53		simplr 792	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. .<_ X )
54		eqid 2622	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
55	18, 22, 54	omndadd 29706	. . . . . . . 8 |- ( ( M e. oMnd /\ ( .0. e. B /\ X e. B /\ ( n .x. X ) e. B ) /\ .0. .<_ X ) -> ( .0. ( +g ` M ) ( n .x. X ) ) .<_ ( X ( +g ` M ) ( n .x. X ) ) )
56	47, 49, 50, 52, 53, 55	syl131anc 1339	. . . . . . 7 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( .0. ( +g ` M ) ( n .x. X ) ) .<_ ( X ( +g ` M ) ( n .x. X ) ) )
57	18, 54, 19	mndlid 17311	. . . . . . . 8 |- ( ( M e. Mnd /\ ( n .x. X ) e. B ) -> ( .0. ( +g ` M ) ( n .x. X ) ) = ( n .x. X ) )
58	48, 52, 57	syl2anc 693	. . . . . . 7 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( .0. ( +g ` M ) ( n .x. X ) ) = ( n .x. X ) )
59	38	a1i 11	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> 1 e. NN0 )
60	18, 26, 54	mulgnn0dir 17571	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( 1 e. NN0 /\ n e. NN0 /\ X e. B ) ) -> ( ( 1 + n ) .x. X ) = ( ( 1 .x. X ) ( +g ` M ) ( n .x. X ) ) )
61	48, 59, 51, 50, 60	syl13anc 1328	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( ( 1 + n ) .x. X ) = ( ( 1 .x. X ) ( +g ` M ) ( n .x. X ) ) )
62		1cnd 10056	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> 1 e. CC )
63		simpr3 1069	. . . . . . . . . . . 12 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. NN0 )
64	63	nn0cnd 11353	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. CC )
65	62, 64	addcomd 10238	. . . . . . . . . 10 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> ( 1 + n ) = ( n + 1 ) )
66	65	3anassrs 1290	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 + n ) = ( n + 1 ) )
67	66	oveq1d 6665	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( ( 1 + n ) .x. X ) = ( ( n + 1 ) .x. X ) )
68	18, 26	mulg1 17548	. . . . . . . . . 10 |- ( X e. B -> ( 1 .x. X ) = X )
69	50, 68	syl 17	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 .x. X ) = X )
70	69	oveq1d 6665	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( ( 1 .x. X ) ( +g ` M ) ( n .x. X ) ) = ( X ( +g ` M ) ( n .x. X ) ) )
71	61, 67, 70	3eqtr3rd 2665	. . . . . . 7 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( X ( +g ` M ) ( n .x. X ) ) = ( ( n + 1 ) .x. X ) )
72	56, 58, 71	3brtr3d 4684	. . . . . 6 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( n .x. X ) .<_ ( ( n + 1 ) .x. X ) )
73	72	adantr 481	. . . . 5 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( n .x. X ) .<_ ( ( n + 1 ) .x. X ) )
74	18, 22	postr 16953	. . . . . 6 |- ( ( M e. Poset /\ ( .0. e. B /\ ( n .x. X ) e. B /\ ( ( n + 1 ) .x. X ) e. B ) ) -> ( ( .0. .<_ ( n .x. X ) /\ ( n .x. X ) .<_ ( ( n + 1 ) .x. X ) ) -> .0. .<_ ( ( n + 1 ) .x. X ) ) )
75	74	imp 445	. . . . 5 |- ( ( ( M e. Poset /\ ( .0. e. B /\ ( n .x. X ) e. B /\ ( ( n + 1 ) .x. X ) e. B ) ) /\ ( .0. .<_ ( n .x. X ) /\ ( n .x. X ) .<_ ( ( n + 1 ) .x. X ) ) ) -> .0. .<_ ( ( n + 1 ) .x. X ) )
76	30, 45, 46, 73, 75	syl22anc 1327	. . . 4 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> .0. .<_ ( ( n + 1 ) .x. X ) )
77	7, 9, 11, 13, 29, 76	nn0indd 11474	. . 3 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ N e. NN0 ) -> .0. .<_ ( N .x. X ) )
78	5, 77	mpdan 702	. 2 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )
79	4, 78	sylbi 207	1 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   lecple 15948   0gc0g 16100   Posetcpo 16940  Tosetctos 17033   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:  omndmul3  29713