Theorem etransclem12 40463

Description: C applied to N. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem12.1	|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
etransclem12.2	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
etransclem12	|- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )

Distinct variable groups:   M, c, n   N, c, n   j, n   ph, n

Allowed substitution hints:   ph( j, c)   C( j, n, c)   M( j)   N( j)

Proof of Theorem etransclem12

Step	Hyp	Ref	Expression
1		etransclem12.1	. . 3 |- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
2	1	a1i 11	. 2 |- ( ph -> C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } ) )
3		oveq2 6658	. . . . 5 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
4	3	oveq1d 6665	. . . 4 |- ( n = N -> ( ( 0 ... n ) ^m ( 0 ... M ) ) = ( ( 0 ... N ) ^m ( 0 ... M ) ) )
5		eqeq2 2633	. . . 4 |- ( n = N -> ( sum_ j e. ( 0 ... M ) ( c ` j ) = n <-> sum_ j e. ( 0 ... M ) ( c ` j ) = N ) )
6	4, 5	rabeqbidv 3195	. . 3 |- ( n = N -> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )
7	6	adantl 482	. 2 |- ( ( ph /\ n = N ) -> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )
8		etransclem12.2	. 2 |- ( ph -> N e. NN0 )
9		ovex 6678	. . . 4 |- ( ( 0 ... N ) ^m ( 0 ... M ) ) e. _V
10	9	rabex 4813	. . 3 |- { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } e. _V
11	10	a1i 11	. 2 |- ( ph -> { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } e. _V )
12	2, 7, 8, 11	fvmptd 6288	1 |- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   NN0cn0 11292   ...cfz 12326   sum_csu 14416

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  etransclem16  40467  etransclem24  40475  etransclem26  40477  etransclem28  40479  etransclem31  40482  etransclem32  40483  etransclem34  40485  etransclem35  40486  etransclem37  40488  etransclem38  40489