Theorem srgpcompp 13362

Description: If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.)

Hypotheses

Ref	Expression
srgpcomp.s	|- S = ( Base ` R )
srgpcomp.m	|- .X. = ( .r ` R )
srgpcomp.g	|- G = (mulGrp ` R )
srgpcomp.e	|- .^ = (.g ` G )
srgpcomp.r	|- ( ph -> R e. SRing )
srgpcomp.a	|- ( ph -> A e. S )
srgpcomp.b	|- ( ph -> B e. S )
srgpcomp.k	|- ( ph -> K e. NN0 )
srgpcomp.c	|- ( ph -> ( A .X. B ) = ( B .X. A ) )
srgpcompp.n	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
srgpcompp	|- ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) )

Proof of Theorem srgpcompp

Step	Hyp	Ref	Expression
1		srgpcomp.r	. . 3 |- ( ph -> R e. SRing )
2		srgpcomp.g	. . . . . . 7 |- G = (mulGrp ` R )
3	2	srgmgp 13339	. . . . . 6 |- ( R e. SRing -> G e. Mnd )
4	1, 3	syl 14	. . . . 5 |- ( ph -> G e. Mnd )
5		srgpcompp.n	. . . . 5 |- ( ph -> N e. NN0 )
6		srgpcomp.a	. . . . . 6 |- ( ph -> A e. S )
7		srgpcomp.s	. . . . . . . 8 |- S = ( Base ` R )
8	2, 7	mgpbasg 13297	. . . . . . 7 |- ( R e. SRing -> S = ( Base ` G ) )
9	1, 8	syl 14	. . . . . 6 |- ( ph -> S = ( Base ` G ) )
10	6, 9	eleqtrd 2268	. . . . 5 |- ( ph -> A e. ( Base ` G ) )
11		eqid 2189	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
12		srgpcomp.e	. . . . . 6 |- .^ = (.g ` G )
13	11, 12	mulgnn0cl 13095	. . . . 5 |- ( ( G e. Mnd /\ N e. NN0 /\ A e. ( Base ` G ) ) -> ( N .^ A ) e. ( Base ` G ) )
14	4, 5, 10, 13	syl3anc 1249	. . . 4 |- ( ph -> ( N .^ A ) e. ( Base ` G ) )
15	14, 9	eleqtrrd 2269	. . 3 |- ( ph -> ( N .^ A ) e. S )
16		srgpcomp.k	. . . . 5 |- ( ph -> K e. NN0 )
17		srgpcomp.b	. . . . . 6 |- ( ph -> B e. S )
18	17, 9	eleqtrd 2268	. . . . 5 |- ( ph -> B e. ( Base ` G ) )
19	11, 12	mulgnn0cl 13095	. . . . 5 |- ( ( G e. Mnd /\ K e. NN0 /\ B e. ( Base ` G ) ) -> ( K .^ B ) e. ( Base ` G ) )
20	4, 16, 18, 19	syl3anc 1249	. . . 4 |- ( ph -> ( K .^ B ) e. ( Base ` G ) )
21	20, 9	eleqtrrd 2269	. . 3 |- ( ph -> ( K .^ B ) e. S )
22		srgpcomp.m	. . . 4 |- .X. = ( .r ` R )
23	7, 22	srgass 13342	. . 3 |- ( ( R e. SRing /\ ( ( N .^ A ) e. S /\ ( K .^ B ) e. S /\ A e. S ) ) -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) )
24	1, 15, 21, 6, 23	syl13anc 1251	. 2 |- ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) )
25		srgpcomp.c	. . . . 5 |- ( ph -> ( A .X. B ) = ( B .X. A ) )
26	7, 22, 2, 12, 1, 6, 17, 16, 25	srgpcomp 13361	. . . 4 |- ( ph -> ( ( K .^ B ) .X. A ) = ( A .X. ( K .^ B ) ) )
27	26	oveq2d 5913	. . 3 |- ( ph -> ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) )
28	7, 22	srgass 13342	. . . 4 |- ( ( R e. SRing /\ ( ( N .^ A ) e. S /\ A e. S /\ ( K .^ B ) e. S ) ) -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) )
29	1, 15, 6, 21, 28	syl13anc 1251	. . 3 |- ( ph -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) )
30	27, 29	eqtr4d 2225	. 2 |- ( ph -> ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) = ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) )
31	2, 22	mgpplusgg 13295	. . . . . 6 |- ( R e. SRing -> .X. = ( +g ` G ) )
32	1, 31	syl 14	. . . . 5 |- ( ph -> .X. = ( +g ` G ) )
33	32	oveqd 5914	. . . 4 |- ( ph -> ( ( N .^ A ) .X. A ) = ( ( N .^ A ) ( +g ` G ) A ) )
34		eqid 2189	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
35	11, 12, 34	mulgnn0p1 13090	. . . . 5 |- ( ( G e. Mnd /\ N e. NN0 /\ A e. ( Base ` G ) ) -> ( ( N + 1 ) .^ A ) = ( ( N .^ A ) ( +g ` G ) A ) )
36	4, 5, 10, 35	syl3anc 1249	. . . 4 |- ( ph -> ( ( N + 1 ) .^ A ) = ( ( N .^ A ) ( +g ` G ) A ) )
37	33, 36	eqtr4d 2225	. . 3 |- ( ph -> ( ( N .^ A ) .X. A ) = ( ( N + 1 ) .^ A ) )
38	37	oveq1d 5912	. 2 |- ( ph -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) )
39	24, 30, 38	3eqtrd 2226	1 |- ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   ` cfv 5235  (class class class)co 5897   1c1 7843   + caddc 7845   NN0cn0 9207   Basecbs 12515   +g cplusg 12592   .rcmulr 12593   Mndcmnd 12892  ._gcmg 13076  mulGrpcmgp 13291  SRingcsrg 13334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-ltadd 7958

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-2 9009  df-3 9010  df-n0 9208  df-z 9285  df-uz 9560  df-seqfrec 10479  df-ndx 12518  df-slot 12519  df-base 12521  df-sets 12522  df-plusg 12605  df-mulr 12606  df-0g 12766  df-mgm 12835  df-sgrp 12880  df-mnd 12893  df-minusg 12964  df-mulg 13077  df-mgp 13292  df-ur 13331  df-srg 13335

This theorem is referenced by:  srgpcomppsc  13363