Theorem gsumnunsn 29418

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 12225	. . 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 17	. 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 11209	. . . 4 |- ( P e. ( ZZ>= ` N ) -> ( P + 1 ) e. ( ZZ>= ` N ) )
8	1, 7	syl 17	. . 3 |- ( ph -> ( P + 1 ) e. ( ZZ>= ` N ) )
9		gsumncl.b	. . . . . 6 |- ( ( ph /\ k e. ( N ... P ) ) -> B e. K )
10	9	adantlr 720	. . . . 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 720	. . . . . 6 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> B = C )
13		gsumnunsn.l	. . . . . . 7 |- ( ph -> C e. K )
14	13	ad2antrr 731	. . . . . 6 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> C e. K )
15	12, 14	eqeltrd 2528	. . . . 5 |- ( ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) /\ k = ( P + 1 ) ) -> B e. K )
16		elfzp1 11843	. . . . . . 7 |- ( P e. ( ZZ>= ` N ) -> ( k e. ( N ... ( P + 1 ) ) <-> ( k e. ( N ... P ) \/ k = ( P + 1 ) ) ) )
17	1, 16	syl 17	. . . . . 6 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) <-> ( k e. ( N ... P ) \/ k = ( P + 1 ) ) ) )
18	17	biimpa 487	. . . . 5 |- ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) -> ( k e. ( N ... P ) \/ k = ( P + 1 ) ) )
19	10, 15, 18	mpjaodan 794	. . . 4 |- ( ( ph /\ k e. ( N ... ( P + 1 ) ) ) -> B e. K )
20		eqid 2450	. . . 4 |- ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B )
21	19, 20	fmptd 6044	. . 3 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) : ( N ... ( P + 1 ) ) --> K )
22	4, 5, 6, 8, 21	gsumval2 16516	. 2 |- ( ph -> ( M gsumg ( k e. ( N ... ( P + 1 ) ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... ( P + 1 ) ) |-> B ) ) ` ( P + 1 ) ) )
23		eqid 2450	. . . . . 6 |- ( k e. ( N ... P ) |-> B ) = ( k e. ( N ... P ) |-> B )
24	9, 23	fmptd 6044	. . . . 5 |- ( ph -> ( k e. ( N ... P ) |-> B ) : ( N ... P ) --> K )
25	4, 5, 6, 1, 24	gsumval2 16516	. . . 4 |- ( ph -> ( M gsumg ( k e. ( N ... P ) |-> B ) ) = ( seq N ( .+ , ( k e. ( N ... P ) |-> B ) ) ` P ) )
26		fzssp1 11838	. . . . . . . 8 |- ( N ... P ) C_ ( N ... ( P + 1 ) )
27		resmpt 5153	. . . . . . . 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 5864	. . . . . 6 |- ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... P ) |-> B ) ` i )
30		fvres 5877	. . . . . . 7 |- ( i e. ( N ... P ) -> ( ( ( k e. ( N ... ( P + 1 ) ) |-> B ) |` ( N ... P ) ) ` i ) = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` i ) )
31	30	adantl 468	. . . . . 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 12234	. . . 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 2451	. . . . 5 |- ( ph -> ( k e. ( N ... ( P + 1 ) ) |-> B ) = ( k e. ( N ... ( P + 1 ) ) |-> B ) )
36		eluzfz2 11804	. . . . . 6 |- ( ( P + 1 ) e. ( ZZ>= ` N ) -> ( P + 1 ) e. ( N ... ( P + 1 ) ) )
37	8, 36	syl 17	. . . . 5 |- ( ph -> ( P + 1 ) e. ( N ... ( P + 1 ) ) )
38	35, 11, 37, 13	fvmptd 5952	. . . 4 |- ( ph -> ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) = C )
39	38	eqcomd 2456	. . 3 |- ( ph -> C = ( ( k e. ( N ... ( P + 1 ) ) |-> B ) ` ( P + 1 ) ) )
40	34, 39	oveq12d 6306	. 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 188   \/ wo 370   /\ wa 371   = wceq 1443   e. wcel 1886   C_ wss 3403   |-> cmpt 4460   |` cres 4835   ` cfv 5581  (class class class)co 6288   1c1 9537   + caddc 9539   ZZ>=cuz 11156   ...cfz 11781   seqcseq 12210   Basecbs 15114   +g cplusg 15183   gsum_g cgsu 15332   Mndcmnd 16528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-seq 12211  df-0g 15333  df-gsum 15334

This theorem is referenced by:  signstfvn  29451