Theorem gsumnunsn 28757

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 12104	. . 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 11135	. . . 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 712	. . . . 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 712	. . . . . 6 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> B = C )
13		gsumnunsn.l	. . . . . . 7 |- ( ph -> C e. K )
14	13	ad2antrr 723	. . . . . 6 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> C e. K )
15	12, 14	eqeltrd 2542	. . . . 5 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> B e. K )
16		elfzp1 11734	. . . . . . 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 482	. . . . 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 2454	. . . 4 |- ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B )
21	19, 20	fmptd 6031	. . 3 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) : ( N ... ( P + 1 ) ) --> K )
22	4, 5, 6, 8, 21	gsumval2 16106	. 2 |- ( ph -> ( M gsumg ( k e. ( N ... ( P + 1 ) ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` ( P + 1 ) ) )
23		eqid 2454	. . . . . 6 |- ( k e. ( N ... P ) |-> B ) = ( k e. ( N ... P ) |-> B )
24	9, 23	fmptd 6031	. . . . 5 |- ( ph -> ( k e. ( N ... P ) |-> B ) : ( N ... P ) --> K )
25	4, 5, 6, 1, 24	gsumval2 16106	. . . 4 |- ( ph -> ( M gsumg ( k e. ( N ... P ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... P ) |-> B ) ) ` P ) )
26		fzssp1 11730	. . . . . . . 8 |- ( N ... P ) C_ ( N ... ( P + 1 ) )
27		resmpt 5311	. . . . . . . 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 5849	. . . . . 6 |- ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... P ) |-> B ) ` i )
30		fvres 5862	. . . . . . 7 |- ( i e. ( N ... P ) -> ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` i ) )
31	30	adantl 464	. . . . . 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 2510	. . . . 5 |- ( ( ph /\ i e. ( N ... P ) ) -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` i ) = ( ( k e. ( N ... P ) |-> B ) ` i ) )
33	1, 32	seqfveq 12113	. . . 4 |- ( ph -> ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) = ( seq N ( .+ , ( k e. ( N ... P ) |-> B ) ) ` P ) )
34	25, 33	eqtr4d 2498	. . 3 |- ( ph -> ( M gsumg ( k e. ( N ... P ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` P ) )
35		eqidd 2455	. . . . 5 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B ) )
36		eluzfz2 11697	. . . . . 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 5936	. . . 4 |- ( ph -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) = C )
39	38	eqcomd 2462	. . 3 |- ( ph -> C = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) )
40	34, 39	oveq12d 6288	. 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 2505	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 366   /\ wa 367   = wceq 1398   e. wcel 1823   C_ wss 3461   |-> cmpt 4497   |` cres 4990   ` cfv 5570  (class class class)co 6270   1c1 9482   + caddc 9484   ZZ>=cuz 11082   ...cfz 11675   seqcseq 12089   Basecbs 14716   +g cplusg 14784   gsum_g cgsu 14930   Mndcmnd 16118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12090  df-0g 14931  df-gsum 14932

This theorem is referenced by:  signstfvn  28790