Theorem gsumprval 13231

Description: Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)

Hypotheses

Ref	Expression
gsumprval.b	|- B = ( Base ` G )
gsumprval.p	|- .+ = ( +g ` G )
gsumprval.g	|- ( ph -> G e. V )
gsumprval.m	|- ( ph -> M e. ZZ )
gsumprval.n	|- ( ph -> N = ( M + 1 ) )
gsumprval.f	|- ( ph -> F : { M , N } --> B )

Assertion

Ref	Expression
gsumprval	|- ( ph -> ( G gsumg F ) = ( ( F ` M ) .+ ( F ` N ) ) )

Proof of Theorem gsumprval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumprval.b	. . 3 |- B = ( Base ` G )
2		gsumprval.p	. . 3 |- .+ = ( +g ` G )
3		gsumprval.g	. . 3 |- ( ph -> G e. V )
4		gsumprval.m	. . . . 5 |- ( ph -> M e. ZZ )
5	4	uzidd 9663	. . . 4 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9704	. . . 4 |- ( M e. ( ZZ>= ` M ) -> ( M + 1 ) e. ( ZZ>= ` M ) )
7	5, 6	syl 14	. . 3 |- ( ph -> ( M + 1 ) e. ( ZZ>= ` M ) )
8		gsumprval.f	. . . 4 |- ( ph -> F : { M , N } --> B )
9		fzpr 10199	. . . . . . 7 |- ( M e. ZZ -> ( M ... ( M + 1 ) ) = { M , ( M + 1 ) } )
10	4, 9	syl 14	. . . . . 6 |- ( ph -> ( M ... ( M + 1 ) ) = { M , ( M + 1 ) } )
11		gsumprval.n	. . . . . . . 8 |- ( ph -> N = ( M + 1 ) )
12	11	eqcomd 2211	. . . . . . 7 |- ( ph -> ( M + 1 ) = N )
13	12	preq2d 3717	. . . . . 6 |- ( ph -> { M , ( M + 1 ) } = { M , N } )
14	10, 13	eqtrd 2238	. . . . 5 |- ( ph -> ( M ... ( M + 1 ) ) = { M , N } )
15	14	feq2d 5413	. . . 4 |- ( ph -> ( F : ( M ... ( M + 1 ) ) --> B <-> F : { M , N } --> B ) )
16	8, 15	mpbird 167	. . 3 |- ( ph -> F : ( M ... ( M + 1 ) ) --> B )
17	1, 2, 3, 7, 16	gsumval2 13229	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( M + 1 ) ) )
18	4	peano2zd 9498	. . . . . . . 8 |- ( ph -> ( M + 1 ) e. ZZ )
19	11, 18	eqeltrd 2282	. . . . . . 7 |- ( ph -> N e. ZZ )
20		prexg 4255	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> { M , N } e. _V )
21	4, 19, 20	syl2anc 411	. . . . . 6 |- ( ph -> { M , N } e. _V )
22	8, 21	fexd 5814	. . . . 5 |- ( ph -> F e. _V )
23		vex 2775	. . . . 5 |- x e. _V
24		fvexg 5595	. . . . 5 |- ( ( F e. _V /\ x e. _V ) -> ( F ` x ) e. _V )
25	22, 23, 24	sylancl 413	. . . 4 |- ( ph -> ( F ` x ) e. _V )
26	25	adantr 276	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. _V )
27		plusgslid 12944	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
28	27	slotex 12859	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
29	3, 28	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
30	2, 29	eqeltrid 2292	. . . . 5 |- ( ph -> .+ e. _V )
31		vex 2775	. . . . . 6 |- y e. _V
32	31	a1i 9	. . . . 5 |- ( ph -> y e. _V )
33		ovexg 5978	. . . . 5 |- ( ( x e. _V /\ .+ e. _V /\ y e. _V ) -> ( x .+ y ) e. _V )
34	23, 30, 32, 33	mp3an2i 1355	. . . 4 |- ( ph -> ( x .+ y ) e. _V )
35	34	adantr 276	. . 3 |- ( ( ph /\ ( x e. _V /\ y e. _V ) ) -> ( x .+ y ) e. _V )
36	5, 26, 35	seq3p1 10610	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( M + 1 ) ) = ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) )
37	4, 26, 35	seq3-1 10607	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
38	12	fveq2d 5580	. . 3 |- ( ph -> ( F ` ( M + 1 ) ) = ( F ` N ) )
39	37, 38	oveq12d 5962	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) = ( ( F ` M ) .+ ( F ` N ) ) )
40	17, 36, 39	3eqtrd 2242	1 |- ( ph -> ( G gsumg F ) = ( ( F ` M ) .+ ( F ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   {cpr 3634   -->wf 5267   ` cfv 5271  (class class class)co 5944   1c1 7926   + caddc 7928   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592   Basecbs 12832   +g cplusg 12909   gsum_g cgsu 13089

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-er 6620  df-en 6828  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-seqfrec 10593  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-igsum 13091

This theorem is referenced by:  gsumpr12val  13232