Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem33 Structured version   Visualization version   GIF version

Theorem etransclem33 45578
Description: 𝐹 is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem33.s (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})
etransclem33.x (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))
etransclem33.p (πœ‘ β†’ 𝑃 ∈ β„•)
etransclem33.m (πœ‘ β†’ 𝑀 ∈ β„•0)
etransclem33.f 𝐹 = (π‘₯ ∈ 𝑋 ↦ ((π‘₯↑(𝑃 βˆ’ 1)) Β· βˆπ‘— ∈ (1...𝑀)((π‘₯ βˆ’ 𝑗)↑𝑃)))
etransclem33.n (πœ‘ β†’ 𝑁 ∈ β„•0)
Assertion
Ref Expression
etransclem33 (πœ‘ β†’ ((𝑆 D𝑛 𝐹)β€˜π‘):π‘‹βŸΆβ„‚)
Distinct variable groups:   𝑗,𝑀,π‘₯   𝑗,𝑁,π‘₯   𝑃,𝑗,π‘₯   𝑆,𝑗,π‘₯   𝑗,𝑋,π‘₯   πœ‘,𝑗,π‘₯
Allowed substitution hints:   𝐹(π‘₯,𝑗)

Proof of Theorem etransclem33
Dummy variables 𝑐 𝑑 π‘˜ π‘š 𝑛 𝑀 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2728 . . . . . . 7 (πœ‘ β†’ (π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š}) = (π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š}))
2 oveq2 7422 . . . . . . . . . 10 (π‘š = 𝑁 β†’ (0...π‘š) = (0...𝑁))
32oveq1d 7429 . . . . . . . . 9 (π‘š = 𝑁 β†’ ((0...π‘š) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀)))
4 eqeq2 2739 . . . . . . . . 9 (π‘š = 𝑁 β†’ (Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š ↔ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁))
53, 4rabeqbidv 3444 . . . . . . . 8 (π‘š = 𝑁 β†’ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š} = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁})
65adantl 481 . . . . . . 7 ((πœ‘ ∧ π‘š = 𝑁) β†’ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š} = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁})
7 etransclem33.n . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ β„•0)
8 ovex 7447 . . . . . . . . 9 ((0...𝑁) ↑m (0...𝑀)) ∈ V
98rabex 5328 . . . . . . . 8 {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁} ∈ V
109a1i 11 . . . . . . 7 (πœ‘ β†’ {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁} ∈ V)
111, 6, 7, 10fvmptd 7006 . . . . . 6 (πœ‘ β†’ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘) = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁})
12 fzfi 13961 . . . . . . . 8 (0...𝑁) ∈ Fin
13 fzfi 13961 . . . . . . . 8 (0...𝑀) ∈ Fin
14 mapfi 9364 . . . . . . . 8 (((0...𝑁) ∈ Fin ∧ (0...𝑀) ∈ Fin) β†’ ((0...𝑁) ↑m (0...𝑀)) ∈ Fin)
1512, 13, 14mp2an 691 . . . . . . 7 ((0...𝑁) ↑m (0...𝑀)) ∈ Fin
16 ssrab2 4073 . . . . . . 7 {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁} βŠ† ((0...𝑁) ↑m (0...𝑀))
17 ssfi 9189 . . . . . . 7 ((((0...𝑁) ↑m (0...𝑀)) ∈ Fin ∧ {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁} βŠ† ((0...𝑁) ↑m (0...𝑀))) β†’ {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁} ∈ Fin)
1815, 16, 17mp2an 691 . . . . . 6 {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁} ∈ Fin
1911, 18eqeltrdi 2836 . . . . 5 (πœ‘ β†’ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘) ∈ Fin)
2019adantr 480 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘) ∈ Fin)
217faccld 14267 . . . . . . . 8 (πœ‘ β†’ (!β€˜π‘) ∈ β„•)
2221nncnd 12250 . . . . . . 7 (πœ‘ β†’ (!β€˜π‘) ∈ β„‚)
2322ad2antrr 725 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ (!β€˜π‘) ∈ β„‚)
2413a1i 11 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ (0...𝑀) ∈ Fin)
25 simpr 484 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘))
2611adantr 480 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘) = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁})
2725, 26eleqtrd 2830 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ 𝑐 ∈ {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = 𝑁})
2816, 27sselid 3976 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)))
29 elmapi 8859 . . . . . . . . . . . . 13 (𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) β†’ 𝑐:(0...𝑀)⟢(0...𝑁))
3028, 29syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ 𝑐:(0...𝑀)⟢(0...𝑁))
3130ffvelcdmda 7088 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ (π‘β€˜π‘—) ∈ (0...𝑁))
3231adantllr 718 . . . . . . . . . 10 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ (π‘β€˜π‘—) ∈ (0...𝑁))
33 elfznn0 13618 . . . . . . . . . 10 ((π‘β€˜π‘—) ∈ (0...𝑁) β†’ (π‘β€˜π‘—) ∈ β„•0)
3432, 33syl 17 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ (π‘β€˜π‘—) ∈ β„•0)
3534faccld 14267 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ (!β€˜(π‘β€˜π‘—)) ∈ β„•)
3635nncnd 12250 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ (!β€˜(π‘β€˜π‘—)) ∈ β„‚)
3724, 36fprodcl 15920 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—)) ∈ β„‚)
3835nnne0d 12284 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ (!β€˜(π‘β€˜π‘—)) β‰  0)
3924, 36, 38fprodn0 15947 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—)) β‰  0)
4023, 37, 39divcld 12012 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ ((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) ∈ β„‚)
41 etransclem33.s . . . . . . . . 9 (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})
4241ad3antrrr 729 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ 𝑆 ∈ {ℝ, β„‚})
43 etransclem33.x . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))
4443ad3antrrr 729 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))
45 etransclem33.p . . . . . . . . 9 (πœ‘ β†’ 𝑃 ∈ β„•)
4645ad3antrrr 729 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ 𝑃 ∈ β„•)
47 etransclem5 45550 . . . . . . . 8 (π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃)))) = (𝑀 ∈ (0...𝑀) ↦ (𝑧 ∈ 𝑋 ↦ ((𝑧 βˆ’ 𝑀)↑if(𝑀 = 0, (𝑃 βˆ’ 1), 𝑃))))
48 simpr 484 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ 𝑗 ∈ (0...𝑀))
4942, 44, 46, 47, 48, 34etransclem20 45565 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ ((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—)):π‘‹βŸΆβ„‚)
50 simpllr 775 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ π‘₯ ∈ 𝑋)
5149, 50ffvelcdmd 7089 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) ∧ 𝑗 ∈ (0...𝑀)) β†’ (((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯) ∈ β„‚)
5224, 51fprodcl 15920 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯) ∈ β„‚)
5340, 52mulcld 11256 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)) β†’ (((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) Β· βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯)) ∈ β„‚)
5420, 53fsumcl 15703 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ Σ𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)(((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) Β· βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯)) ∈ β„‚)
55 eqid 2727 . . 3 (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)(((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) Β· βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯))) = (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)(((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) Β· βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯)))
5654, 55fmptd 7118 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)(((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) Β· βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯))):π‘‹βŸΆβ„‚)
57 etransclem33.m . . . 4 (πœ‘ β†’ 𝑀 ∈ β„•0)
58 etransclem33.f . . . 4 𝐹 = (π‘₯ ∈ 𝑋 ↦ ((π‘₯↑(𝑃 βˆ’ 1)) Β· βˆπ‘— ∈ (1...𝑀)((π‘₯ βˆ’ 𝑗)↑𝑃)))
59 etransclem5 45550 . . . 4 (π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (π‘₯ ∈ 𝑋 ↦ ((π‘₯ βˆ’ 𝑗)↑if(𝑗 = 0, (𝑃 βˆ’ 1), 𝑃))))
60 etransclem11 45556 . . . 4 (π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š}) = (𝑛 ∈ β„•0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(π‘β€˜π‘—) = 𝑛})
6141, 43, 45, 57, 58, 7, 59, 60etransclem30 45575 . . 3 (πœ‘ β†’ ((𝑆 D𝑛 𝐹)β€˜π‘) = (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)(((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) Β· βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯))))
6261feq1d 6701 . 2 (πœ‘ β†’ (((𝑆 D𝑛 𝐹)β€˜π‘):π‘‹βŸΆβ„‚ ↔ (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ ((π‘š ∈ β„•0 ↦ {𝑑 ∈ ((0...π‘š) ↑m (0...𝑀)) ∣ Ξ£π‘˜ ∈ (0...𝑀)(π‘‘β€˜π‘˜) = π‘š})β€˜π‘)(((!β€˜π‘) / βˆπ‘— ∈ (0...𝑀)(!β€˜(π‘β€˜π‘—))) Β· βˆπ‘— ∈ (0...𝑀)(((𝑆 D𝑛 ((π‘˜ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 βˆ’ π‘˜)↑if(π‘˜ = 0, (𝑃 βˆ’ 1), 𝑃))))β€˜π‘—))β€˜(π‘β€˜π‘—))β€˜π‘₯))):π‘‹βŸΆβ„‚))
6356, 62mpbird 257 1 (πœ‘ β†’ ((𝑆 D𝑛 𝐹)β€˜π‘):π‘‹βŸΆβ„‚)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3427  Vcvv 3469   βŠ† wss 3944  ifcif 4524  {cpr 4626   ↦ cmpt 5225  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ↑m cmap 8836  Fincfn 8955  β„‚cc 11128  β„cr 11129  0cc0 11130  1c1 11131   Β· cmul 11135   βˆ’ cmin 11466   / cdiv 11893  β„•cn 12234  β„•0cn0 12494  ...cfz 13508  β†‘cexp 14050  !cfa 14256  Ξ£csu 15656  βˆcprod 15873   β†Ύt crest 17393  TopOpenctopn 17394  β„‚fldccnfld 21266   D𝑛 cdvn 25780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208  ax-addf 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8838  df-pm 8839  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9378  df-fi 9426  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-q 12955  df-rp 12999  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-ico 13354  df-icc 13355  df-fz 13509  df-fzo 13652  df-seq 13991  df-exp 14051  df-fac 14257  df-bc 14286  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-prod 15874  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-starv 17239  df-sca 17240  df-vsca 17241  df-ip 17242  df-tset 17243  df-ple 17244  df-ds 17246  df-unif 17247  df-hom 17248  df-cco 17249  df-rest 17395  df-topn 17396  df-0g 17414  df-gsum 17415  df-topgen 17416  df-pt 17417  df-prds 17420  df-xrs 17475  df-qtop 17480  df-imas 17481  df-xps 17483  df-mre 17557  df-mrc 17558  df-acs 17560  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-submnd 18732  df-mulg 19015  df-cntz 19259  df-cmn 19728  df-psmet 21258  df-xmet 21259  df-met 21260  df-bl 21261  df-mopn 21262  df-fbas 21263  df-fg 21264  df-cnfld 21267  df-top 22783  df-topon 22800  df-topsp 22822  df-bases 22836  df-cld 22910  df-ntr 22911  df-cls 22912  df-nei 22989  df-lp 23027  df-perf 23028  df-cn 23118  df-cnp 23119  df-haus 23206  df-tx 23453  df-hmeo 23646  df-fil 23737  df-fm 23829  df-flim 23830  df-flf 23831  df-xms 24213  df-ms 24214  df-tms 24215  df-cncf 24785  df-limc 25782  df-dv 25783  df-dvn 25784
This theorem is referenced by:  etransclem39  45584  etransclem43  45588  etransclem45  45590  etransclem46  45591  etransclem47  45592
  Copyright terms: Public domain W3C validator