Theorem omndmul2 28482

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 988	. . 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 659	. . . 4 |- ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) <-> ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) ) )
3	2	anbi1i 706	. . 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 260	. 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 767	. . 3 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> N e. NN0 )
6		oveq1 6283	. . . . 5 |- ( m = 0 -> ( m .x. X ) = ( 0 .x. X ) )
7	6	breq2d 4386	. . . 4 |- ( m = 0 -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( 0 .x. X ) ) )
8		oveq1 6283	. . . . 5 |- ( m = n -> ( m .x. X ) = ( n .x. X ) )
9	8	breq2d 4386	. . . 4 |- ( m = n -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( n .x. X ) ) )
10		oveq1 6283	. . . . 5 |- ( m = ( n + 1 ) -> ( m .x. X ) = ( ( n + 1 ) .x. X ) )
11	10	breq2d 4386	. . . 4 |- ( m = ( n + 1 ) -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( ( n + 1 ) .x. X ) ) )
12		oveq1 6283	. . . . 5 |- ( m = N -> ( m .x. X ) = ( N .x. X ) )
13	12	breq2d 4386	. . . 4 |- ( m = N -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( N .x. X ) ) )
14		omndtos 28475	. . . . . . . 8 |- ( M e. oMnd -> M e. Toset )
15		tospos 28427	. . . . . . . 8 |- ( M e. Toset -> M e. Poset )
16	14, 15	syl 17	. . . . . . 7 |- ( M e. oMnd -> M e. Poset )
17		omndmnd 28474	. . . . . . . 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 16565	. . . . . . . 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 16207	. . . . . . 7 |- ( ( M e. Poset /\ .0. e. B ) -> .0. .<_ .0. )
24	16, 21, 23	syl2anc 671	. . . . . 6 |- ( M e. oMnd -> .0. .<_ .0. )
25	24	ad3antrrr 741	. . . . 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 16774	. . . . . 6 |- ( X e. B -> ( 0 .x. X ) = .0. )
28	27	ad3antlr 742	. . . . 5 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> ( 0 .x. X ) = .0. )
29	25, 28	breqtrrd 4401	. . . 4 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( 0 .x. X ) )
30	16	ad5antr 745	. . . . 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 745	. . . . . . 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 767	. . . . . . 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 784	. . . . . . 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 16785	. . . . . . 7 |- ( ( M e. Mnd /\ n e. NN0 /\ X e. B ) -> ( n .x. X ) e. B )
36	31, 33, 34, 35	syl3anc 1271	. . . . . 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 1100	. . . . . . . . . 10 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> n e. NN0 )
38		1nn0 10875	. . . . . . . . . . 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 10919	. . . . . . . . 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 1235	. . . . . . . 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 1235	. . . . . . 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 16785	. . . . . . 7 |- ( ( M e. Mnd /\ ( n + 1 ) e. NN0 /\ X e. B ) -> ( ( n + 1 ) .x. X ) e. B )
44	31, 42, 34, 43	syl3anc 1271	. . . . . 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 1189	. . . . 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 467	. . . . 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 781	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. oMnd )
48	17	ad4antr 743	. . . . . . . . 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 782	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> X e. B )
51		simpr 467	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> n e. NN0 )
52	48, 51, 50, 35	syl3anc 1271	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( n .x. X ) e. B )
53		simplr 767	. . . . . . . 8 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. .<_ X )
54		eqid 2452	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
55	18, 22, 54	omndadd 28476	. . . . . . . 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 1284	. . . . . . 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 16568	. . . . . . . 8 |- ( ( M e. Mnd /\ ( n .x. X ) e. B ) -> ( .0. ( +g ` M ) ( n .x. X ) ) = ( n .x. X ) )
58	48, 52, 57	syl2anc 671	. . . . . . 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 16792	. . . . . . . . 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 1273	. . . . . . . 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 9646	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> 1 e. CC )
63		simpr3 1017	. . . . . . . . . . . 12 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. NN0 )
64	63	nn0cnd 10917	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. CC )
65	62, 64	addcomd 9822	. . . . . . . . . 10 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> ( 1 + n ) = ( n + 1 ) )
66	65	3anassrs 1235	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 + n ) = ( n + 1 ) )
67	66	oveq1d 6291	. . . . . . . 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 16776	. . . . . . . . . 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 6291	. . . . . . . 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 2495	. . . . . . 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 4404	. . . . . 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 471	. . . . 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 16210	. . . . . 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 435	. . . . 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 1272	. . . 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 11022	. . 3 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ N e. NN0 ) -> .0. .<_ ( N .x. X ) )
78	5, 77	mpdan 679	. 2 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )
79	4, 78	sylbi 200	1 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( N .x. X ) )

Syntax hints:   -> wi 4   /\ wa 375   /\ w3a 986   = wceq 1448   e. wcel 1891   class class class wbr 4374   ` cfv 5561  (class class class)co 6276   0cc0 9526   1c1 9527   + caddc 9529   NN0cn0 10859   Basecbs 15132   +g cplusg 15201   lecple 15208   0gc0g 15349   Posetcpo 16196  Tosetctos 16290   Mndcmnd 16546  ._gcmg 16683  oMndcomnd 28467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-inf2 8133  ax-cnex 9582  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-1st 6781  df-2nd 6782  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-nn 10599  df-n0 10860  df-z 10928  df-uz 11150  df-fz 11776  df-seq 12208  df-0g 15351  df-preset 16184  df-poset 16202  df-toset 16291  df-mgm 16499  df-sgrp 16538  df-mnd 16548  df-mulg 16687  df-omnd 28469

This theorem is referenced by:  omndmul3  28483