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Mirrors > Home > ILE Home > Th. List > iisermulc2 | GIF version |
Description: Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
clim2ser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isermulc2.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isermulc2.4 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
isermulc2.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴) |
isermulc2.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
isermulc2.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
Ref | Expression |
---|---|
iisermulc2 | ⊢ (𝜑 → seq𝑀( + , 𝐺, ℂ) ⇝ (𝐶 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2ser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isermulc2.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isermulc2.5 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴) | |
4 | isermulc2.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | iseqex 9740 | . . 3 ⊢ seq𝑀( + , 𝐺, ℂ) ∈ V | |
6 | 5 | a1i 9 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐺, ℂ) ∈ V) |
7 | isermulc2.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
8 | 1, 2, 7 | iserf 9766 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ) |
9 | 8 | ffvelrnda 5377 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ∈ ℂ) |
10 | addcl 7368 | . . . 4 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
11 | 10 | adantl 271 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
12 | 4 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℂ) |
13 | adddi 7375 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) | |
14 | 13 | 3expb 1140 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) |
15 | 12, 14 | sylan 277 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) |
16 | simpr 108 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
17 | 16, 1 | syl6eleq 2175 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
18 | 1 | eleq2i 2149 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀)) |
19 | 18, 7 | sylan2br 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
20 | 19 | adantlr 461 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
21 | isermulc2.7 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) | |
22 | 18, 21 | sylan2br 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
23 | 22 | adantlr 461 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
24 | cnex 7367 | . . . 4 ⊢ ℂ ∈ V | |
25 | 24 | a1i 9 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ℂ ∈ V) |
26 | mulcl 7370 | . . . 4 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ) | |
27 | 26 | adantl 271 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ) |
28 | 4 | adantr 270 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℂ) |
29 | 28, 7 | mulcld 7409 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐶 · (𝐹‘𝑘)) ∈ ℂ) |
30 | 21, 29 | eqeltrd 2159 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
31 | 18, 30 | sylan2br 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ) |
32 | 31 | adantlr 461 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ) |
33 | 11, 15, 17, 20, 23, 25, 27, 32, 12 | iseqdistr 9784 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺, ℂ)‘𝑗) = (𝐶 · (seq𝑀( + , 𝐹, ℂ)‘𝑗))) |
34 | 1, 2, 3, 4, 6, 9, 33 | climmulc2 10541 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐺, ℂ) ⇝ (𝐶 · 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 Vcvv 2612 class class class wbr 3811 ‘cfv 4967 (class class class)co 5589 ℂcc 7249 + caddc 7254 · cmul 7256 ℤcz 8644 ℤ≥cuz 8912 seqcseq 9738 ⇝ cli 10489 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 ax-arch 7365 ax-caucvg 7366 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-2 8373 df-3 8374 df-4 8375 df-n0 8564 df-z 8645 df-uz 8913 df-rp 9028 df-iseq 9739 df-iexp 9790 df-cj 10101 df-re 10102 df-im 10103 df-rsqrt 10256 df-abs 10257 df-clim 10490 |
This theorem is referenced by: (None) |
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