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Theorem etransclem13 44478
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 2736 . . 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 2736 . . 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 44469 . 2 (𝜑𝐹 = (𝑥𝑋 ↦ ∏𝑗 ∈ (0...𝑀)(((𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))‘𝑗)‘𝑥)))
8 simpr 485 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
9 cnex 11132 . . . . . . . . 9 ℂ ∈ V
109ssex 5278 . . . . . . . 8 (𝑋 ⊆ ℂ → 𝑋 ∈ V)
11 mptexg 7171 . . . . . . . 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 481 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
14 oveq1 7364 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
1514oveq1d 7372 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1615cbvmptv 5218 . . . . . . . 8 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
1716mpteq2i 5210 . . . . . . 7 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
1817fvmpt2 6959 . . . . . 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 584 . . . . 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 713 . . . 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 485 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑥)
22 simpl 483 . . . . . . . 8 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑥 = 𝑌)
2321, 22eqtrd 2776 . . . . . . 7 ((𝑥 = 𝑌𝑦 = 𝑥) → 𝑦 = 𝑌)
24 oveq1 7364 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
2524oveq1d 7372 . . . . . . 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 712 . . . . 5 (((𝜑𝑥 = 𝑌) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
2827adantlr 713 . . . 4 ((((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑦 = 𝑥) → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
29 simpr 485 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
30 etransclem13.y . . . . . . 7 (𝜑𝑌𝑋)
3130adantr 481 . . . . . 6 ((𝜑𝑥 = 𝑌) → 𝑌𝑋)
3229, 31eqeltrd 2838 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥𝑋)
3332adantr 481 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
34 ovexd 7392 . . . 4 (((𝜑𝑥 = 𝑌) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
3520, 28, 33, 34fvmptd 6955 . . 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 15806 . 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 15790 . . 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 6955 1 (𝜑 → (𝐹𝑌) = ∏𝑗 ∈ (0...𝑀)((𝑌𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  wss 3910  ifcif 4486  cmpt 5188  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051  1c1 11052   · cmul 11056  cmin 11385  cn 12153  0cn0 12413  ...cfz 13424  cexp 13967  cprod 15788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-prod 15789
This theorem is referenced by:  etransclem18  44483  etransclem23  44488  etransclem46  44511
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