Theorem omndmul2 26108

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 962	. . 3 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) /\ .0. .<_ X ) <-> ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) /\ .0. .<_ X ) )
2		df-3an 962	. . . . 5 |- ( ( M e. oMnd /\ X e. B /\ N e. NN0 ) <-> ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) )
3		3anass 964	. . . . 5 |- ( ( M e. oMnd /\ X e. B /\ N e. NN0 ) <-> ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) )
4	2, 3	bitr3i 251	. . . 4 |- ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) <-> ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) )
5	4	anbi1i 690	. . 3 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) <-> ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) /\ .0. .<_ X ) )
6	1, 5	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 ) )
7		simplr 749	. . 3 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> N e. NN0 )
8		oveq1 6097	. . . . 5 |- ( m = 0 -> ( m .x. X ) = ( 0 .x. X ) )
9	8	breq2d 4301	. . . 4 |- ( m = 0 -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( 0 .x. X ) ) )
10		oveq1 6097	. . . . 5 |- ( m = n -> ( m .x. X ) = ( n .x. X ) )
11	10	breq2d 4301	. . . 4 |- ( m = n -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( n .x. X ) ) )
12		oveq1 6097	. . . . 5 |- ( m = ( n + 1 ) -> ( m .x. X ) = ( ( n + 1 ) .x. X ) )
13	12	breq2d 4301	. . . 4 |- ( m = ( n + 1 ) -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( ( n + 1 ) .x. X ) ) )
14		oveq1 6097	. . . . 5 |- ( m = N -> ( m .x. X ) = ( N .x. X ) )
15	14	breq2d 4301	. . . 4 |- ( m = N -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( N .x. X ) ) )
16		omndtos 26101	. . . . . . . 8 |- ( M e. oMnd -> M e. Toset )
17		tospos 26052	. . . . . . . 8 |- ( M e. Toset -> M e. Poset )
18	16, 17	syl 16	. . . . . . 7 |- ( M e. oMnd -> M e. Poset )
19		omndmnd 26100	. . . . . . . 8 |- ( M e. oMnd -> M e. Mnd )
20		omndmul.0	. . . . . . . . 9 |- B = ( Base ` M )
21		omndmul2.3	. . . . . . . . 9 |- .0. = ( 0g ` M )
22	20, 21	mndidcl 15435	. . . . . . . 8 |- ( M e. Mnd -> .0. e. B )
23	19, 22	syl 16	. . . . . . 7 |- ( M e. oMnd -> .0. e. B )
24		omndmul.1	. . . . . . . 8 |- .<_ = ( le ` M )
25	20, 24	posref 15117	. . . . . . 7 |- ( ( M e. Poset /\ .0. e. B ) -> .0. .<_ .0. )
26	18, 23, 25	syl2anc 656	. . . . . 6 |- ( M e. oMnd -> .0. .<_ .0. )
27	26	ad3antrrr 724	. . . . 5 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ .0. )
28		omndmul2.2	. . . . . . 7 |- .x. = (.g ` M )
29	20, 21, 28	mulg0 15625	. . . . . 6 |- ( X e. B -> ( 0 .x. X ) = .0. )
30	29	ad3antlr 725	. . . . 5 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> ( 0 .x. X ) = .0. )
31	27, 30	breqtrrd 4315	. . . 4 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( 0 .x. X ) )
32	18	ad5antr 728	. . . . 5 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> M e. Poset )
33	19	ad5antr 728	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> M e. Mnd )
34	33, 22	syl 16	. . . . . 6 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> .0. e. B )
35		simplr 749	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> n e. NN0 )
36		simp-5r 763	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> X e. B )
37	20, 28	mulgnn0cl 15636	. . . . . . 7 |- ( ( M e. Mnd /\ n e. NN0 /\ X e. B ) -> ( n .x. X ) e. B )
38	33, 35, 36, 37	syl3anc 1213	. . . . . 6 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( n .x. X ) e. B )
39		simpr32 1074	. . . . . . . . . 10 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> n e. NN0 )
40		1nn0 10591	. . . . . . . . . . 11 |- 1 e. NN0
41	40	a1i 11	. . . . . . . . . 10 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> 1 e. NN0 )
42	39, 41	nn0addcld 10636	. . . . . . . . 9 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> ( n + 1 ) e. NN0 )
43	42	3anassrs 1204	. . . . . . . 8 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) -> ( n + 1 ) e. NN0 )
44	43	3anassrs 1204	. . . . . . 7 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( n + 1 ) e. NN0 )
45	20, 28	mulgnn0cl 15636	. . . . . . 7 |- ( ( M e. Mnd /\ ( n + 1 ) e. NN0 /\ X e. B ) -> ( ( n + 1 ) .x. X ) e. B )
46	33, 44, 36, 45	syl3anc 1213	. . . . . 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 )
47	34, 38, 46	3jca 1163	. . . . 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 ) )
48		simpr 458	. . . . . 6 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> .0. .<_ ( n .x. X ) )
49		simp-4l 760	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. oMnd )
50	19	ad4antr 726	. . . . . . . . . 10 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. Mnd )
51	50, 22	syl 16	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. e. B )
52		simp-4r 761	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> X e. B )
53		simpr 458	. . . . . . . . . 10 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> n e. NN0 )
54	50, 53, 52, 37	syl3anc 1213	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( n .x. X ) e. B )
55		simplr 749	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. .<_ X )
56		eqid 2441	. . . . . . . . . 10 |- ( +g ` M ) = ( +g ` M )
57	20, 24, 56	omndadd 26102	. . . . . . . . 9 |- ( ( 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 ) ) )
58	49, 51, 52, 54, 55, 57	syl131anc 1226	. . . . . . . 8 |- ( ( ( ( ( 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 ) ) )
59	20, 56, 21	mndlid 15437	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( n .x. X ) e. B ) -> ( .0. ( +g ` M ) ( n .x. X ) ) = ( n .x. X ) )
60	50, 54, 59	syl2anc 656	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( .0. ( +g ` M ) ( n .x. X ) ) = ( n .x. X ) )
61	40	a1i 11	. . . . . . . . . 10 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> 1 e. NN0 )
62	20, 28, 56	mulgnn0dir 15643	. . . . . . . . . 10 |- ( ( M e. Mnd /\ ( 1 e. NN0 /\ n e. NN0 /\ X e. B ) ) -> ( ( 1 + n ) .x. X ) = ( ( 1 .x. X ) ( +g ` M ) ( n .x. X ) ) )
63	50, 61, 53, 52, 62	syl13anc 1215	. . . . . . . . 9 |- ( ( ( ( ( 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 ) ) )
64		ax-1cn 9336	. . . . . . . . . . . . 13 |- 1 e. CC
65	64	a1i 11	. . . . . . . . . . . 12 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> 1 e. CC )
66		simpr3 991	. . . . . . . . . . . . 13 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. NN0 )
67	66	nn0cnd 10634	. . . . . . . . . . . 12 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. CC )
68	65, 67	addcomd 9567	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> ( 1 + n ) = ( n + 1 ) )
69	68	3anassrs 1204	. . . . . . . . . 10 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 + n ) = ( n + 1 ) )
70	69	oveq1d 6105	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( ( 1 + n ) .x. X ) = ( ( n + 1 ) .x. X ) )
71	20, 28	mulg1 15627	. . . . . . . . . . 11 |- ( X e. B -> ( 1 .x. X ) = X )
72	52, 71	syl 16	. . . . . . . . . 10 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 .x. X ) = X )
73	72	oveq1d 6105	. . . . . . . . 9 |- ( ( ( ( ( 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 ) ) )
74	63, 70, 73	3eqtr3rd 2482	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( X ( +g ` M ) ( n .x. X ) ) = ( ( n + 1 ) .x. X ) )
75	58, 60, 74	3brtr3d 4318	. . . . . . 7 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( n .x. X ) .<_ ( ( n + 1 ) .x. X ) )
76	75	adantr 462	. . . . . 6 |- ( ( ( ( ( ( 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 ) )
77	48, 76	jca 529	. . . . 5 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> ( .0. .<_ ( n .x. X ) /\ ( n .x. X ) .<_ ( ( n + 1 ) .x. X ) ) )
78	20, 24	postr 15119	. . . . . 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 ) ) )
79	78	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 ) )
80	32, 47, 77, 79	syl21anc 1212	. . . 4 |- ( ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) /\ .0. .<_ ( n .x. X ) ) -> .0. .<_ ( ( n + 1 ) .x. X ) )
81	9, 11, 13, 15, 31, 80	nn0indd 26021	. . 3 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ N e. NN0 ) -> .0. .<_ ( N .x. X ) )
82	7, 81	mpdan 663	. 2 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )
83	6, 82	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 960   = wceq 1364   e. wcel 1761   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279   + caddc 9281   NN0cn0 10575   Basecbs 14170   +g cplusg 14234   lecple 14241   0gc0g 14374   Posetcpo 15106  Tosetctos 15199   Mndcmnd 15405  ._gcmg 15410  oMndcomnd 26093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-0g 14376  df-poset 15112  df-toset 15200  df-mnd 15411  df-mulg 15541  df-omnd 26095

This theorem is referenced by:  omndmul3  26109