Theorem gsumnunsn 28366

Description: Closure of a group sum in a non-commutative monoid. (Contributed by Thierry Arnoux, 8-Oct-2018.)

Hypotheses

Ref	Expression
gsumncl.k	|- K = ( Base ` M )
gsumncl.w	|- ( ph -> M e. Mnd )
gsumncl.p	|- ( ph -> P e. ( ZZ>= ` N ) )
gsumncl.b	|- ( ( ph /\ k e. ( N ... P ) ) -> B e. K )
gsumnunsn.a	|- .+ = ( +g ` M )
gsumnunsn.l	|- ( ph -> C e. K )
gsumnunsn.c	|- ( ( ph /\ k = ( P + 1 ) ) -> B = C )

Assertion

Ref	Expression
gsumnunsn	|- ( ph -> ( M gsumg ( k e. ( N ... ( P + 1 ) ) |-> B ) ) = ( ( M gsumg ( k e. ( N ... P ) |-> B ) ) .+ C ) )

Distinct variable groups:   k, K   k, N   P, k   ph, k   C, k

Allowed substitution hints:   B( k)   .+ ( k)   M( k)

Proof of Theorem gsumnunsn

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumncl.p	. . 3 |- ( ph -> P e. ( ZZ>= ` N ) )
2		seqp1 12101	. . 3 |- ( P e. ( ZZ>= ` N ) -> ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` ( P + 1 ) ) = ( ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) .+ ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) ) )
3	1, 2	syl 16	. 2 |- ( ph -> ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` ( P + 1 ) ) = ( ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) .+ ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) ) )
4		gsumncl.k	. . 3 |- K = ( Base ` M )
5		gsumnunsn.a	. . 3 |- .+ = ( +g ` M )
6		gsumncl.w	. . 3 |- ( ph -> M e. Mnd )
7		peano2uz 11143	. . . 4 |- ( P e. ( ZZ>= ` N ) -> ( P + 1 ) e. ( ZZ>= ` N ) )
8	1, 7	syl 16	. . 3 |- ( ph -> ( P + 1 ) e. ( ZZ>= ` N ) )
9		gsumncl.b	. . . . . 6 |- ( ( ph /\ k e. ( N ... P ) ) -> B e. K )
10	9	adantlr 714	. . . . 5 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k e. ( N ... P ) ) -> B e. K )
11		gsumnunsn.c	. . . . . . 7 |- ( ( ph /\ k = ( P + 1 ) ) -> B = C )
12	11	adantlr 714	. . . . . 6 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> B = C )
13		gsumnunsn.l	. . . . . . 7 |- ( ph -> C e. K )
14	13	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> C e. K )
15	12, 14	eqeltrd 2531	. . . . 5 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> B e. K )
16		elfzp1 11739	. . . . . . 7 |- ( P e. ( ZZ>= ` N ) -> ( k e. ( N ... ( P + 1 ) ) <-> ( k e. ( N ... P ) \/ k = ( P + 1 ) ) ) )
17	1, 16	syl 16	. . . . . 6 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) <-> ( k e. ( N ... P ) \/ k = ( P + 1 ) ) ) )
18	17	biimpa 484	. . . . 5 |- ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) -> ( k e. ( N ... P ) \/ k = ( P + 1 ) ) )
19	10, 15, 18	mpjaodan 786	. . . 4 |- ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) -> B e. K )
20		eqid 2443	. . . 4 |- ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B )
21	19, 20	fmptd 6040	. . 3 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) : ( N ... ( P + 1 ) ) --> K )
22	4, 5, 6, 8, 21	gsumval2 15781	. 2 |- ( ph -> ( M gsumg ( k e. ( N ... ( P + 1 ) ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` ( P + 1 ) ) )
23		eqid 2443	. . . . . 6 |- ( k e. ( N ... P ) |-> B ) = ( k e. ( N ... P ) |-> B )
24	9, 23	fmptd 6040	. . . . 5 |- ( ph -> ( k e. ( N ... P ) |-> B ) : ( N ... P ) --> K )
25	4, 5, 6, 1, 24	gsumval2 15781	. . . 4 |- ( ph -> ( M gsumg ( k e. ( N ... P ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... P ) |-> B ) ) ` P ) )
26		fzssp1 11735	. . . . . . . 8 |- ( N ... P ) C_ ( N ... ( P + 1 ) )
27		resmpt 5313	. . . . . . . 8 |- ( ( N ... P ) C_ ( N ... ( P + 1 ) ) -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) = ( k e. ( N ... P ) |-> B ) )
28	26, 27	ax-mp 5	. . . . . . 7 |- ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) = ( k e. ( N ... P ) |-> B )
29	28	fveq1i 5857	. . . . . 6 |- ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... P ) |-> B ) ` i )
30		fvres 5870	. . . . . . 7 |- ( i e. ( N ... P ) -> ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` i ) )
31	30	adantl 466	. . . . . 6 |- ( ( ph /\ i e. ( N ... P ) ) -> ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` i ) )
32	29, 31	syl5reqr 2499	. . . . 5 |- ( ( ph /\ i e. ( N ... P ) ) -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` i ) = ( ( k e. ( N ... P ) |-> B ) ` i ) )
33	1, 32	seqfveq 12110	. . . 4 |- ( ph -> ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) = ( seq N ( .+ , ( k e. ( N ... P ) |-> B ) ) ` P ) )
34	25, 33	eqtr4d 2487	. . 3 |- ( ph -> ( M gsumg ( k e. ( N ... P ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) )
35		eqidd 2444	. . . . 5 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B ) )
36		eluzfz2 11703	. . . . . 6 |- ( ( P + 1 ) e. ( ZZ>= ` N ) -> ( P + 1 ) e. ( N ... ( P + 1 ) ) )
37	8, 36	syl 16	. . . . 5 |- ( ph -> ( P + 1 ) e. ( N ... ( P + 1 ) ) )
38	35, 11, 37, 13	fvmptd 5946	. . . 4 |- ( ph -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) = C )
39	38	eqcomd 2451	. . 3 |- ( ph -> C = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) )
40	34, 39	oveq12d 6299	. 2 |- ( ph -> ( ( M gsumg ( k e. ( N ... P ) |-> B ) ) .+ C ) = ( ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) .+ ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) ) )
41	3, 22, 40	3eqtr4d 2494	1 |- ( ph -> ( M gsumg ( k e. ( N ... ( P + 1 ) ) |-> B ) ) = ( ( M gsumg ( k e. ( N ... P ) |-> B ) ) .+ C ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1383   e. wcel 1804   C_ wss 3461   |-> cmpt 4495   |` cres 4991   ` cfv 5578  (class class class)co 6281   1c1 9496   + caddc 9498   ZZ>=cuz 11090   ...cfz 11681   seqcseq 12086   Basecbs 14509   +g cplusg 14574   gsum_g cgsu 14715   Mndcmnd 15793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-seq 12087  df-0g 14716  df-gsum 14717

This theorem is referenced by:  signstfvn  28399