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Theorem etransclem13 40967
Description: 𝐹 applied to 𝑌. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem13.x (𝜑𝑋 ⊆ ℂ)
etransclem13.p (𝜑𝑃 ∈ ℕ)
etransclem13.m (𝜑𝑀 ∈ ℕ0)
etransclem13.f 𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))
etransclem13.y (𝜑𝑌𝑋)
Assertion
Ref Expression
etransclem13 (𝜑 → (𝐹𝑌) = ∏𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
Distinct variable groups:   𝑗,𝑀,𝑥   𝑃,𝑗,𝑥   𝑗,𝑋,𝑥   𝑗,𝑌,𝑥   𝜑,𝑗,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑗)

Proof of Theorem etransclem13
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 etransclem13.x . . 3 (𝜑𝑋 ⊆ ℂ)
2 etransclem13.p . . 3 (𝜑𝑃 ∈ ℕ)
3 etransclem13.m . . 3 (𝜑𝑀 ∈ ℕ0)
4 etransclem13.f . . 3 𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))
5 eqid 2760 . . 3 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
6 eqid 2760 . . 3 (𝑥𝑋 ↦ ∏𝑗 ∈ (0...𝑀)(((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥)) = (𝑥𝑋 ↦ ∏𝑗 ∈ (0...𝑀)(((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥))
71, 2, 3, 4, 5, 6etransclem4 40958 . 2 (𝜑𝐹 = (𝑥𝑋 ↦ ∏𝑗 ∈ (0...𝑀)(((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥)))
8 simpr 479 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
9 cnex 10209 . . . . . . . . 9 ℂ ∈ V
109ssex 4954 . . . . . . . 8 (𝑋 ⊆ ℂ → 𝑋 ∈ V)
11 mptexg 6648 . . . . . . . 8 (𝑋 ∈ V → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
121, 10, 113syl 18 . . . . . . 7 (𝜑 → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
1312adantr 472 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
14 oveq1 6820 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
1514oveq1d 6828 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1615cbvmptv 4902 . . . . . . . 8 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1716mpteq2i 4893 . . . . . . 7 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
1817fvmpt2 6453 . . . . . 6 ((𝑗 ∈ (0...𝑀) ∧ (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V) → ((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
198, 13, 18syl2anc 696 . . . . 5 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
2019adantlr 753 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
21 simpr 479 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑥)
22 simpl 474 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑥 = 𝑌)
2321, 22eqtrd 2794 . . . . . . 7 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑌)
24 oveq1 6820 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
2524oveq1d 6828 . . . . . . 7 (𝑦 = 𝑌 → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
2623, 25syl 17 . . . . . 6 ((𝑥 = 𝑌𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
2726adantll 752 . . . . 5 (((𝜑𝑥 = 𝑌) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
2827adantlr 753 . . . 4 ((((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
29 simpr 479 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
30 etransclem13.y . . . . . . 7 (𝜑𝑌𝑋)
3130adantr 472 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑌𝑋)
3229, 31eqeltrd 2839 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥𝑋)
3332adantr 472 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
34 ovexd 6843 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
3520, 28, 33, 34fvmptd 6450 . . 3 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
3635prodeq2dv 14852 . 2 ((𝜑𝑥 = 𝑌) → ∏𝑗 ∈ (0...𝑀)(((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥) = ∏𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
37 prodex 14836 . . 3 𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V
3837a1i 11 . 2 (𝜑 → ∏𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
397, 36, 30, 38fvmptd 6450 1 (𝜑 → (𝐹𝑌) = ∏𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715  ifcif 4230  cmpt 4881  cfv 6049  (class class class)co 6813  cc 10126  0cc0 10128  1c1 10129   · cmul 10133  cmin 10458  cn 11212  0cn0 11484  ...cfz 12519  cexp 13054  cprod 14834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-prod 14835
This theorem is referenced by:  etransclem18  40972  etransclem23  40977  etransclem46  41000
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