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Theorem etransclem13 38941
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 2606 . . 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 2606 . . 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 38932 . 2 (𝜑𝐹 = (𝑥𝑋 ↦ ∏𝑗 ∈ (0...𝑀)(((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥)))
8 simpr 475 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
9 cnex 9870 . . . . . . . . 9 ℂ ∈ V
109ssex 4722 . . . . . . . 8 (𝑋 ⊆ ℂ → 𝑋 ∈ V)
11 mptexg 6364 . . . . . . . 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 479 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
14 oveq1 6531 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
1514oveq1d 6539 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1615cbvmptv 4669 . . . . . . . 8 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1716mpteq2i 4660 . . . . . . 7 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
1817fvmpt2 6182 . . . . . 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 690 . . . . 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 746 . . . 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 475 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑥)
22 simpl 471 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑥 = 𝑌)
2321, 22eqtrd 2640 . . . . . . 7 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑌)
24 oveq1 6531 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
2524oveq1d 6539 . . . . . . 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 745 . . . . 5 (((𝜑𝑥 = 𝑌) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
2827adantlr 746 . . . 4 ((((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
29 simpr 475 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
30 etransclem13.y . . . . . . 7 (𝜑𝑌𝑋)
3130adantr 479 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑌𝑋)
3229, 31eqeltrd 2684 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥𝑋)
3332adantr 479 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
34 ovex 6552 . . . . 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 6179 . . 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 14435 . 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 14419 . . 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 6179 1 (𝜑 → (𝐹𝑌) = ∏𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3169  wss 3536  ifcif 4032  cmpt 4634  cfv 5787  (class class class)co 6524  cc 9787  0cc0 9789  1c1 9790   · cmul 9794  cmin 10114  cn 10864  0cn0 11136  ...cfz 12149  cexp 12674  cprod 14417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-oi 8272  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-fz 12150  df-fzo 12287  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-clim 14010  df-prod 14418
This theorem is referenced by:  etransclem18  38946  etransclem23  38951  etransclem46  38974
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