Theorem omndmul2 27862

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 975	. . 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 649	. . . 4 |- ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) <-> ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) )
3	2	anbi1i 695	. . 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 252	. 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 755	. . 3 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> N e. NN0 )
6		oveq1 6303	. . . . 5 |- ( m = 0 -> ( m .x. X ) = ( 0 .x. X ) )
7	6	breq2d 4468	. . . 4 |- ( m = 0 -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( 0 .x. X ) ) )
8		oveq1 6303	. . . . 5 |- ( m = n -> ( m .x. X ) = ( n .x. X ) )
9	8	breq2d 4468	. . . 4 |- ( m = n -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( n .x. X ) ) )
10		oveq1 6303	. . . . 5 |- ( m = ( n + 1 ) -> ( m .x. X ) = ( ( n + 1 ) .x. X ) )
11	10	breq2d 4468	. . . 4 |- ( m = ( n + 1 ) -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( ( n + 1 ) .x. X ) ) )
12		oveq1 6303	. . . . 5 |- ( m = N -> ( m .x. X ) = ( N .x. X ) )
13	12	breq2d 4468	. . . 4 |- ( m = N -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( N .x. X ) ) )
14		omndtos 27855	. . . . . . . 8 |- ( M e. oMnd -> M e. Toset )
15		tospos 27806	. . . . . . . 8 |- ( M e. Toset -> M e. Poset )
16	14, 15	syl 16	. . . . . . 7 |- ( M e. oMnd -> M e. Poset )
17		omndmnd 27854	. . . . . . . 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 16065	. . . . . . . 8 |- ( M e. Mnd -> .0. e. B )
21	17, 20	syl 16	. . . . . . 7 |- ( M e. oMnd -> .0. e. B )
22		omndmul.1	. . . . . . . 8 |- .<_ = ( le ` M )
23	18, 22	posref 15707	. . . . . . 7 |- ( ( M e. Poset /\ .0. e. B ) -> .0. .<_ .0. )
24	16, 21, 23	syl2anc 661	. . . . . 6 |- ( M e. oMnd -> .0. .<_ .0. )
25	24	ad3antrrr 729	. . . . 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 16274	. . . . . 6 |- ( X e. B -> ( 0 .x. X ) = .0. )
28	27	ad3antlr 730	. . . . 5 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> ( 0 .x. X ) = .0. )
29	25, 28	breqtrrd 4482	. . . 4 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( 0 .x. X ) )
30	16	ad5antr 733	. . . . 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 733	. . . . . . 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 16	. . . . . 6 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> .0. e. B )
33		simplr 755	. . . . . . 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 770	. . . . . . 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 16285	. . . . . . 7 |- ( ( M e. Mnd /\ n e. NN0 /\ X e. B ) -> ( n .x. X ) e. B )
36	31, 33, 34, 35	syl3anc 1228	. . . . . 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 1087	. . . . . . . . . 10 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> n e. NN0 )
38		1nn0 10832	. . . . . . . . . . 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 10877	. . . . . . . . 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 1218	. . . . . . . 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 1218	. . . . . . 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 16285	. . . . . . 7 |- ( ( M e. Mnd /\ ( n + 1 ) e. NN0 /\ X e. B ) -> ( ( n + 1 ) .x. X ) e. B )
44	31, 42, 34, 43	syl3anc 1228	. . . . . 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 1176	. . . . 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 461	. . . . 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 767	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. oMnd )
48	17	ad4antr 731	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. Mnd )
49	48, 20	syl 16	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. e. B )
50		simp-4r 768	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> X e. B )
51		simpr 461	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> n e. NN0 )
52	48, 51, 50, 35	syl3anc 1228	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( n .x. X ) e. B )
53		simplr 755	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. .<_ X )
54		eqid 2457	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
55	18, 22, 54	omndadd 27856	. . . . . . . 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 1241	. . . . . . 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 16068	. . . . . . . 8 |- ( ( M e. Mnd /\ ( n .x. X ) e. B ) -> ( .0. ( +g ` M ) ( n .x. X ) ) = ( n .x. X ) )
58	48, 52, 57	syl2anc 661	. . . . . . 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 16292	. . . . . . . . 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 1230	. . . . . . . 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 9629	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> 1 e. CC )
63		simpr3 1004	. . . . . . . . . . . 12 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. NN0 )
64	63	nn0cnd 10875	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. CC )
65	62, 64	addcomd 9799	. . . . . . . . . 10 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> ( 1 + n ) = ( n + 1 ) )
66	65	3anassrs 1218	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 + n ) = ( n + 1 ) )
67	66	oveq1d 6311	. . . . . . . 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 16276	. . . . . . . . . 10 |- ( X e. B -> ( 1 .x. X ) = X )
69	50, 68	syl 16	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 .x. X ) = X )
70	69	oveq1d 6311	. . . . . . . 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 2507	. . . . . . 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 4485	. . . . . 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 465	. . . . 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 15710	. . . . . 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 429	. . . . 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 1229	. . . 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 10982	. . 3 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ N e. NN0 ) -> .0. .<_ ( N .x. X ) )
78	5, 77	mpdan 668	. 2 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )
79	4, 78	sylbi 195	1 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   + caddc 9512   NN0cn0 10816   Basecbs 14644   +g cplusg 14712   lecple 14719   0gc0g 14857   Posetcpo 15696  Tosetctos 15790   Mndcmnd 16046  ._gcmg 16183  oMndcomnd 27847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111  df-0g 14859  df-preset 15684  df-poset 15702  df-toset 15791  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mulg 16187  df-omnd 27849

This theorem is referenced by:  omndmul3  27863