Theorem reprval 30688

Description: Value of the representations of M as the sum of S nonnegative integers in a given set A (Contributed by Thierry Arnoux, 1-Dec-2021.)

Hypotheses

Ref	Expression
reprval.a	|- ( ph -> A C_ NN )
reprval.m	|- ( ph -> M e. ZZ )
reprval.s	|- ( ph -> S e. NN0 )

Assertion

Ref	Expression
reprval	|- ( ph -> ( A (repr ` S ) M ) = { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } )

Distinct variable groups:   A, c   M, c   S, a, c   ph, c

Allowed substitution hints:   ph( a)   A( a)   M( a)

Proof of Theorem reprval

Dummy variables b m s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-repr 30687	. . . 4 |- repr = ( s e. NN0 |-> ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ s ) ) | sum_ a e. ( 0..^ s ) ( c ` a ) = m } ) )
2	1	a1i 11	. . 3 |- ( ph -> repr = ( s e. NN0 |-> ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ s ) ) | sum_ a e. ( 0..^ s ) ( c ` a ) = m } ) ) )
3		oveq2 6658	. . . . . . 7 |- ( s = S -> ( 0..^ s ) = ( 0..^ S ) )
4	3	oveq2d 6666	. . . . . 6 |- ( s = S -> ( b ^m ( 0..^ s ) ) = ( b ^m ( 0..^ S ) ) )
5	3	sumeq1d 14431	. . . . . . 7 |- ( s = S -> sum_ a e. ( 0..^ s ) ( c ` a ) = sum_ a e. ( 0..^ S ) ( c ` a ) )
6	5	eqeq1d 2624	. . . . . 6 |- ( s = S -> ( sum_ a e. ( 0..^ s ) ( c ` a ) = m <-> sum_ a e. ( 0..^ S ) ( c ` a ) = m ) )
7	4, 6	rabeqbidv 3195	. . . . 5 |- ( s = S -> { c e. ( b ^m ( 0..^ s ) ) | sum_ a e. ( 0..^ s ) ( c ` a ) = m } = { c e. ( b ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = m } )
8	7	mpt2eq3dv 6721	. . . 4 |- ( s = S -> ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ s ) ) | sum_ a e. ( 0..^ s ) ( c ` a ) = m } ) = ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = m } ) )
9	8	adantl 482	. . 3 |- ( ( ph /\ s = S ) -> ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ s ) ) | sum_ a e. ( 0..^ s ) ( c ` a ) = m } ) = ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = m } ) )
10		reprval.s	. . 3 |- ( ph -> S e. NN0 )
11		nnex 11026	. . . . . 6 |- NN e. _V
12	11	pwex 4848	. . . . 5 |- ~P NN e. _V
13		zex 11386	. . . . 5 |- ZZ e. _V
14	12, 13	mpt2ex 7247	. . . 4 |- ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = m } ) e. _V
15	14	a1i 11	. . 3 |- ( ph -> ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = m } ) e. _V )
16	2, 9, 10, 15	fvmptd 6288	. 2 |- ( ph -> (repr ` S ) = ( b e. ~P NN , m e. ZZ |-> { c e. ( b ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = m } ) )
17		simprl 794	. . . 4 |- ( ( ph /\ ( b = A /\ m = M ) ) -> b = A )
18	17	oveq1d 6665	. . 3 |- ( ( ph /\ ( b = A /\ m = M ) ) -> ( b ^m ( 0..^ S ) ) = ( A ^m ( 0..^ S ) ) )
19		simprr 796	. . . 4 |- ( ( ph /\ ( b = A /\ m = M ) ) -> m = M )
20	19	eqeq2d 2632	. . 3 |- ( ( ph /\ ( b = A /\ m = M ) ) -> ( sum_ a e. ( 0..^ S ) ( c ` a ) = m <-> sum_ a e. ( 0..^ S ) ( c ` a ) = M ) )
21	18, 20	rabeqbidv 3195	. 2 |- ( ( ph /\ ( b = A /\ m = M ) ) -> { c e. ( b ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = m } = { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } )
22	11	a1i 11	. . . 4 |- ( ph -> NN e. _V )
23		reprval.a	. . . 4 |- ( ph -> A C_ NN )
24	22, 23	ssexd 4805	. . 3 |- ( ph -> A e. _V )
25	24, 23	elpwd 4167	. 2 |- ( ph -> A e. ~P NN )
26		reprval.m	. 2 |- ( ph -> M e. ZZ )
27		ovex 6678	. . . 4 |- ( A ^m ( 0..^ S ) ) e. _V
28	27	rabex 4813	. . 3 |- { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } e. _V
29	28	a1i 11	. 2 |- ( ph -> { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } e. _V )
30	16, 21, 25, 26, 29	ovmpt2d 6788	1 |- ( ph -> ( A (repr ` S ) M ) = { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   0cc0 9936   NNcn 11020   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   sum_csu 14416  reprcrepr 30686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-neg 10269  df-nn 11021  df-z 11378  df-seq 12802  df-sum 14417  df-repr 30687

This theorem is referenced by:  repr0  30689  reprf  30690  reprsum  30691  reprsuc  30693  reprfi  30694  reprss  30695  reprinrn  30696  reprlt  30697  reprgt  30699  reprinfz1  30700  reprpmtf1o  30704  reprdifc  30705  breprexplema  30708