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

Theorem etransclem13 39140
 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 2610 . . 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 2610 . . 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 39131 . 2 (𝜑𝐹 = (𝑥𝑋 ↦ ∏𝑗 ∈ (0...𝑀)(((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥)))
8 simpr 476 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
9 cnex 9896 . . . . . . . . 9 ℂ ∈ V
109ssex 4730 . . . . . . . 8 (𝑋 ⊆ ℂ → 𝑋 ∈ V)
11 mptexg 6389 . . . . . . . 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 480 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
14 oveq1 6556 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
1514oveq1d 6564 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1615cbvmptv 4678 . . . . . . . 8 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1716mpteq2i 4669 . . . . . . 7 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
1817fvmpt2 6200 . . . . . 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 691 . . . . 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 747 . . . 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 476 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑥)
22 simpl 472 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑥 = 𝑌)
2321, 22eqtrd 2644 . . . . . . 7 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑌)
24 oveq1 6556 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
2524oveq1d 6564 . . . . . . 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 746 . . . . 5 (((𝜑𝑥 = 𝑌) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
2827adantlr 747 . . . 4 ((((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
29 simpr 476 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
30 etransclem13.y . . . . . . 7 (𝜑𝑌𝑋)
3130adantr 480 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑌𝑋)
3229, 31eqeltrd 2688 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥𝑋)
3332adantr 480 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
34 ovex 6577 . . . . 5 ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V
3534a1i 11 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
3620, 28, 33, 35fvmptd 6197 . . 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)))
3736prodeq2dv 14492 . 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)))
38 prodex 14476 . . 3 𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V
3938a1i 11 . 2 (𝜑 → ∏𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
407, 37, 30, 39fvmptd 6197 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 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ifcif 4036   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   · cmul 9820   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ...cfz 12197  ↑cexp 12722  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-prod 14475 This theorem is referenced by:  etransclem18  39145  etransclem23  39150  etransclem46  39173
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