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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem1 | Structured version Visualization version GIF version |
Description: 𝐻 is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem1.x | ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
etransclem1.p | ⊢ (𝜑 → 𝑃 ∈ ℕ) |
etransclem1.h | ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) |
etransclem1.j | ⊢ (𝜑 → 𝐽 ∈ (0...𝑀)) |
Ref | Expression |
---|---|
etransclem1 | ⊢ (𝜑 → (𝐻‘𝐽):𝑋⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ) | |
2 | 1 | sselda 3995 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ) |
3 | etransclem1.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀)) | |
4 | 3 | elfzelzd 13562 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
5 | 4 | zcnd 12721 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ ℂ) |
7 | 2, 6 | subcld 11618 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝐽) ∈ ℂ) |
8 | etransclem1.p | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) | |
9 | nnm1nn0 12565 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0) |
11 | 8 | nnnn0d 12585 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ0) |
12 | 10, 11 | ifcld 4577 | . . . . 5 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0) |
14 | 7, 13 | expcld 14183 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ) |
15 | eqid 2735 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃))) = (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃))) | |
16 | 14, 15 | fmptd 7134 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃))):𝑋⟶ℂ) |
17 | etransclem1.h | . . . . 5 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) | |
18 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑗 = 𝑛 → (𝑥 − 𝑗) = (𝑥 − 𝑛)) | |
19 | eqeq1 2739 | . . . . . . . . 9 ⊢ (𝑗 = 𝑛 → (𝑗 = 0 ↔ 𝑛 = 0)) | |
20 | 19 | ifbid 4554 | . . . . . . . 8 ⊢ (𝑗 = 𝑛 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑛 = 0, (𝑃 − 1), 𝑃)) |
21 | 18, 20 | oveq12d 7449 | . . . . . . 7 ⊢ (𝑗 = 𝑛 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 𝑛)↑if(𝑛 = 0, (𝑃 − 1), 𝑃))) |
22 | 21 | mpteq2dv 5250 | . . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑛)↑if(𝑛 = 0, (𝑃 − 1), 𝑃)))) |
23 | 22 | cbvmptv 5261 | . . . . 5 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑛 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑛)↑if(𝑛 = 0, (𝑃 − 1), 𝑃)))) |
24 | 17, 23 | eqtri 2763 | . . . 4 ⊢ 𝐻 = (𝑛 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑛)↑if(𝑛 = 0, (𝑃 − 1), 𝑃)))) |
25 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 𝐽 → (𝑥 − 𝑛) = (𝑥 − 𝐽)) | |
26 | eqeq1 2739 | . . . . . . 7 ⊢ (𝑛 = 𝐽 → (𝑛 = 0 ↔ 𝐽 = 0)) | |
27 | 26 | ifbid 4554 | . . . . . 6 ⊢ (𝑛 = 𝐽 → if(𝑛 = 0, (𝑃 − 1), 𝑃) = if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
28 | 25, 27 | oveq12d 7449 | . . . . 5 ⊢ (𝑛 = 𝐽 → ((𝑥 − 𝑛)↑if(𝑛 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃))) |
29 | 28 | mpteq2dv 5250 | . . . 4 ⊢ (𝑛 = 𝐽 → (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑛)↑if(𝑛 = 0, (𝑃 − 1), 𝑃))) = (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃)))) |
30 | cnex 11234 | . . . . . 6 ⊢ ℂ ∈ V | |
31 | 30 | ssex 5327 | . . . . 5 ⊢ (𝑋 ⊆ ℂ → 𝑋 ∈ V) |
32 | mptexg 7241 | . . . . 5 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃))) ∈ V) | |
33 | 1, 31, 32 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃))) ∈ V) |
34 | 24, 29, 3, 33 | fvmptd3 7039 | . . 3 ⊢ (𝜑 → (𝐻‘𝐽) = (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃)))) |
35 | 34 | feq1d 6721 | . 2 ⊢ (𝜑 → ((𝐻‘𝐽):𝑋⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝐽)↑if(𝐽 = 0, (𝑃 − 1), 𝑃))):𝑋⟶ℂ)) |
36 | 16, 35 | mpbird 257 | 1 ⊢ (𝜑 → (𝐻‘𝐽):𝑋⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ifcif 4531 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 ...cfz 13544 ↑cexp 14099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-seq 14040 df-exp 14100 |
This theorem is referenced by: etransclem29 46219 |
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