Theorem omndmul2 26315

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 967	. . 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 967	. . . . 5 |- ( ( M e. oMnd /\ X e. B /\ N e. NN0 ) <-> ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) )
3		3anass 969	. . . . 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 695	. . 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 754	. . 3 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> N e. NN0 )
8		oveq1 6202	. . . . 5 |- ( m = 0 -> ( m .x. X ) = ( 0 .x. X ) )
9	8	breq2d 4407	. . . 4 |- ( m = 0 -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( 0 .x. X ) ) )
10		oveq1 6202	. . . . 5 |- ( m = n -> ( m .x. X ) = ( n .x. X ) )
11	10	breq2d 4407	. . . 4 |- ( m = n -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( n .x. X ) ) )
12		oveq1 6202	. . . . 5 |- ( m = ( n + 1 ) -> ( m .x. X ) = ( ( n + 1 ) .x. X ) )
13	12	breq2d 4407	. . . 4 |- ( m = ( n + 1 ) -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( ( n + 1 ) .x. X ) ) )
14		oveq1 6202	. . . . 5 |- ( m = N -> ( m .x. X ) = ( N .x. X ) )
15	14	breq2d 4407	. . . 4 |- ( m = N -> ( .0. .<_ ( m .x. X ) <-> .0. .<_ ( N .x. X ) ) )
16		omndtos 26308	. . . . . . . 8 |- ( M e. oMnd -> M e. Toset )
17		tospos 26259	. . . . . . . 8 |- ( M e. Toset -> M e. Poset )
18	16, 17	syl 16	. . . . . . 7 |- ( M e. oMnd -> M e. Poset )
19		omndmnd 26307	. . . . . . . 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 15553	. . . . . . . 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 15235	. . . . . . 7 |- ( ( M e. Poset /\ .0. e. B ) -> .0. .<_ .0. )
26	18, 23, 25	syl2anc 661	. . . . . 6 |- ( M e. oMnd -> .0. .<_ .0. )
27	26	ad3antrrr 729	. . . . 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 15746	. . . . . 6 |- ( X e. B -> ( 0 .x. X ) = .0. )
30	29	ad3antlr 730	. . . . 5 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> ( 0 .x. X ) = .0. )
31	27, 30	breqtrrd 4421	. . . 4 |- ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( 0 .x. X ) )
32	18	ad5antr 733	. . . . 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 733	. . . . . . 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 754	. . . . . . 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 768	. . . . . . 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 15757	. . . . . . 7 |- ( ( M e. Mnd /\ n e. NN0 /\ X e. B ) -> ( n .x. X ) e. B )
38	33, 35, 36, 37	syl3anc 1219	. . . . . 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 1079	. . . . . . . . . 10 |- ( ( M e. oMnd /\ ( X e. B /\ N e. NN0 /\ ( .0. .<_ X /\ n e. NN0 /\ .0. .<_ ( n .x. X ) ) ) ) -> n e. NN0 )
40		1nn0 10701	. . . . . . . . . . 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 10746	. . . . . . . . 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 1210	. . . . . . . 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 1210	. . . . . . 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 15757	. . . . . . 7 |- ( ( M e. Mnd /\ ( n + 1 ) e. NN0 /\ X e. B ) -> ( ( n + 1 ) .x. X ) e. B )
46	33, 44, 36, 45	syl3anc 1219	. . . . . 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 1168	. . . . 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 461	. . . . . 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 765	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> M e. oMnd )
50	19	ad4antr 731	. . . . . . . . . 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 766	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> X e. B )
53		simpr 461	. . . . . . . . . 10 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> n e. NN0 )
54	50, 53, 52, 37	syl3anc 1219	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( n .x. X ) e. B )
55		simplr 754	. . . . . . . . 9 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> .0. .<_ X )
56		eqid 2452	. . . . . . . . . 10 |- ( +g ` M ) = ( +g ` M )
57	20, 24, 56	omndadd 26309	. . . . . . . . 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 1232	. . . . . . . 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 15555	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( n .x. X ) e. B ) -> ( .0. ( +g ` M ) ( n .x. X ) ) = ( n .x. X ) )
60	50, 54, 59	syl2anc 661	. . . . . . . 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 15764	. . . . . . . . . 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 1221	. . . . . . . . 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 9446	. . . . . . . . . . . . 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 996	. . . . . . . . . . . . 13 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. NN0 )
67	66	nn0cnd 10744	. . . . . . . . . . . 12 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> n e. CC )
68	65, 67	addcomd 9677	. . . . . . . . . . 11 |- ( ( ( M e. oMnd /\ X e. B ) /\ ( N e. NN0 /\ .0. .<_ X /\ n e. NN0 ) ) -> ( 1 + n ) = ( n + 1 ) )
69	68	3anassrs 1210	. . . . . . . . . 10 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ n e. NN0 ) -> ( 1 + n ) = ( n + 1 ) )
70	69	oveq1d 6210	. . . . . . . . 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 15748	. . . . . . . . . . 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 6210	. . . . . . . . 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 2502	. . . . . . . 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 4424	. . . . . . 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 465	. . . . . 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 532	. . . . 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 15237	. . . . . 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 1218	. . . 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 26228	. . 3 |- ( ( ( ( ( M e. oMnd /\ X e. B ) /\ N e. NN0 ) /\ .0. .<_ X ) /\ N e. NN0 ) -> .0. .<_ ( N .x. X ) )
82	7, 81	mpdan 668	. 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 965   = wceq 1370   e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   CCcc 9386   0cc0 9388   1c1 9389   + caddc 9391   NN0cn0 10685   Basecbs 14287   +g cplusg 14352   lecple 14359   0gc0g 14492   Posetcpo 15224  Tosetctos 15317   Mndcmnd 15523  ._gcmg 15528  oMndcomnd 26300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-seq 11919  df-0g 14494  df-poset 15230  df-toset 15318  df-mnd 15529  df-mulg 15662  df-omnd 26302

This theorem is referenced by:  omndmul3  26316