Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem13 Structured version   Visualization version   GIF version

Theorem etransclem13 40884
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 2724 . . 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 2724 . . 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 40875 . 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 10130 . . . . . . . . 9 ℂ ∈ V
109ssex 4910 . . . . . . . 8 (𝑋 ⊆ ℂ → 𝑋 ∈ V)
11 mptexg 6600 . . . . . . . 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 6772 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
1514oveq1d 6780 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1615cbvmptv 4858 . . . . . . . 8 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1716mpteq2i 4849 . . . . . . 7 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
1817fvmpt2 6405 . . . . . 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 2758 . . . . . . 7 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑌)
24 oveq1 6772 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
2524oveq1d 6780 . . . . . . 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 2803 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥𝑋)
3332adantr 472 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
34 ovexd 6795 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
3520, 28, 33, 34fvmptd 6402 . . 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 14773 . 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 14757 . . 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 6402 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 1596  wcel 2103  Vcvv 3304  wss 3680  ifcif 4194  cmpt 4837  cfv 6001  (class class class)co 6765  cc 10047  0cc0 10049  1c1 10050   · cmul 10054  cmin 10379  cn 11133  0cn0 11405  ...cfz 12440  cexp 12975  cprod 14755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-sup 8464  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-rp 11947  df-fz 12441  df-fzo 12581  df-seq 12917  df-exp 12976  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339  df-prod 14756
This theorem is referenced by:  etransclem18  40889  etransclem23  40894  etransclem46  40917
  Copyright terms: Public domain W3C validator