Theorem k0004val 38448

Description: The topological simplex of dimension N is the set of real vectors where the components are nonnegative and sum to 1. (Contributed by RP, 29-Mar-2021.)

Hypothesis

Ref	Expression
k0004.a	|- A = ( n e. NN0 |-> { t e. ( ( 0 [,] 1 ) ^m ( 1 ... ( n + 1 ) ) ) | sum_ k e. ( 1 ... ( n + 1 ) ) ( t ` k ) = 1 } )

Assertion

Ref	Expression
k0004val	|- ( N e. NN0 -> ( A ` N ) = { t e. ( ( 0 [,] 1 ) ^m ( 1 ... ( N + 1 ) ) ) | sum_ k e. ( 1 ... ( N + 1 ) ) ( t ` k ) = 1 } )

Distinct variable groups:   k, n   t, n   k, N   t, N, n

Allowed substitution hints:   A( t, k, n)

Proof of Theorem k0004val

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 |- ( n = N -> ( n + 1 ) = ( N + 1 ) )
2	1	oveq2d 6666	. . . 4 |- ( n = N -> ( 1 ... ( n + 1 ) ) = ( 1 ... ( N + 1 ) ) )
3	2	oveq2d 6666	. . 3 |- ( n = N -> ( ( 0 [,] 1 ) ^m ( 1 ... ( n + 1 ) ) ) = ( ( 0 [,] 1 ) ^m ( 1 ... ( N + 1 ) ) ) )
4	2	sumeq1d 14431	. . . 4 |- ( n = N -> sum_ k e. ( 1 ... ( n + 1 ) ) ( t ` k ) = sum_ k e. ( 1 ... ( N + 1 ) ) ( t ` k ) )
5	4	eqeq1d 2624	. . 3 |- ( n = N -> ( sum_ k e. ( 1 ... ( n + 1 ) ) ( t ` k ) = 1 <-> sum_ k e. ( 1 ... ( N + 1 ) ) ( t ` k ) = 1 ) )
6	3, 5	rabeqbidv 3195	. 2 |- ( n = N -> { t e. ( ( 0 [,] 1 ) ^m ( 1 ... ( n + 1 ) ) ) | sum_ k e. ( 1 ... ( n + 1 ) ) ( t ` k ) = 1 } = { t e. ( ( 0 [,] 1 ) ^m ( 1 ... ( N + 1 ) ) ) | sum_ k e. ( 1 ... ( N + 1 ) ) ( t ` k ) = 1 } )
7		k0004.a	. 2 |- A = ( n e. NN0 |-> { t e. ( ( 0 [,] 1 ) ^m ( 1 ... ( n + 1 ) ) ) | sum_ k e. ( 1 ... ( n + 1 ) ) ( t ` k ) = 1 } )
8		ovex 6678	. . 3 |- ( ( 0 [,] 1 ) ^m ( 1 ... ( N + 1 ) ) ) e. _V
9	8	rabex 4813	. 2 |- { t e. ( ( 0 [,] 1 ) ^m ( 1 ... ( N + 1 ) ) ) | sum_ k e. ( 1 ... ( N + 1 ) ) ( t ` k ) = 1 } e. _V
10	6, 7, 9	fvmpt 6282	1 |- ( N e. NN0 -> ( A ` N ) = { t e. ( ( 0 [,] 1 ) ^m ( 1 ... ( N + 1 ) ) ) | sum_ k e. ( 1 ... ( N + 1 ) ) ( t ` k ) = 1 } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   [,]cicc 12178   ...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-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-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  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-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-sum 14417

This theorem is referenced by:  k0004ss1  38449  k0004val0  38452