Theorem gsumnunsn 27071

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 11922	. . 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 11009	. . . 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 2539	. . . . 5 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> B e. K )
16		elfzp1 11606	. . . . . . 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 784	. . . 4 |- ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) -> B e. K )
20		eqid 2451	. . . 4 |- ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B )
21	19, 20	fmptd 5966	. . 3 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) : ( N ... ( P + 1 ) ) --> K )
22	4, 5, 6, 8, 21	gsumval2 15615	. 2 |- ( ph -> ( M gsumg ( k e. ( N ... ( P + 1 ) ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` ( P + 1 ) ) )
23		eqid 2451	. . . . . 6 |- ( k e. ( N ... P ) |-> B ) = ( k e. ( N ... P ) |-> B )
24	9, 23	fmptd 5966	. . . . 5 |- ( ph -> ( k e. ( N ... P ) |-> B ) : ( N ... P ) --> K )
25	4, 5, 6, 1, 24	gsumval2 15615	. . . 4 |- ( ph -> ( M gsumg ( k e. ( N ... P ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... P ) |-> B ) ) ` P ) )
26		fzssp1 11602	. . . . . . . 8 |- ( N ... P ) C_ ( N ... ( P + 1 ) )
27		resmpt 5254	. . . . . . . 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 5790	. . . . . 6 |- ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... P ) |-> B ) ` i )
30		fvres 5803	. . . . . . 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 2507	. . . . 5 |- ( ( ph /\ i e. ( N ... P ) ) -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` i ) = ( ( k e. ( N ... P ) |-> B ) ` i ) )
33	1, 32	seqfveq 11931	. . . 4 |- ( ph -> ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) = ( seq N ( .+ , ( k e. ( N ... P ) |-> B ) ) ` P ) )
34	25, 33	eqtr4d 2495	. . 3 |- ( ph -> ( M gsumg ( k e. ( N ... P ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) )
35	20	a1i 11	. . . . 5 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B ) )
36		eluzfz2 11560	. . . . . 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 5878	. . . 4 |- ( ph -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) = C )
39	38	eqcomd 2459	. . 3 |- ( ph -> C = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) )
40	34, 39	oveq12d 6208	. 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 2502	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 1370   e. wcel 1758   C_ wss 3426   |-> cmpt 4448   |` cres 4940   ` cfv 5516  (class class class)co 6190   1c1 9384   + caddc 9386   ZZ>=cuz 10962   ...cfz 11538   seqcseq 11907   Basecbs 14276   +g cplusg 14340   gsum_g cgsu 14481   Mndcmnd 15511

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 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460

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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-seq 11908  df-0g 14482  df-gsum 14483

This theorem is referenced by:  signstfvn  27104