Theorem gsumprval 13662

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 9887	. . . 4 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9933	. . . 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 10433	. . . . . . 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 2240	. . . . . . 7 |- ( ph -> ( M + 1 ) = N )
13	12	preq2d 3780	. . . . . 6 |- ( ph -> { M , ( M + 1 ) } = { M , N } )
14	10, 13	eqtrd 2267	. . . . 5 |- ( ph -> ( M ... ( M + 1 ) ) = { M , N } )
15	14	feq2d 5501	. . . 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 13660	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( M + 1 ) ) )
18	4	peano2zd 9721	. . . . . . . 8 |- ( ph -> ( M + 1 ) e. ZZ )
19	11, 18	eqeltrd 2311	. . . . . . 7 |- ( ph -> N e. ZZ )
20		prexg 4330	. . . . . . 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 5921	. . . . 5 |- ( ph -> F e. _V )
23		vex 2818	. . . . 5 |- x e. _V
24		fvexg 5694	. . . . 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 13409	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
28	27	slotex 13323	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
29	3, 28	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
30	2, 29	eqeltrid 2321	. . . . 5 |- ( ph -> .+ e. _V )
31		vex 2818	. . . . . 6 |- y e. _V
32	31	a1i 9	. . . . 5 |- ( ph -> y e. _V )
33		ovexg 6092	. . . . 5 |- ( ( x e. _V /\ .+ e. _V /\ y e. _V ) -> ( x .+ y ) e. _V )
34	23, 30, 32, 33	mp3an2i 1379	. . . 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 10851	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( M + 1 ) ) = ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) )
37	4, 26, 35	seq3-1 10848	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
38	12	fveq2d 5679	. . 3 |- ( ph -> ( F ` ( M + 1 ) ) = ( F ` N ) )
39	37, 38	oveq12d 6076	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) = ( ( F ` M ) .+ ( F ` N ) ) )
40	17, 36, 39	3eqtrd 2271	1 |- ( ph -> ( G gsumg F ) = ( ( F ` M ) .+ ( F ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   {cpr 3695   -->wf 5353   ` cfv 5357  (class class class)co 6058   1c1 8144   + caddc 8146   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   +g cplusg 13374   gsum_g cgsu 13554

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556

This theorem is referenced by:  gsumpr12val  13663