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Mirrors > Home > MPE Home > Th. List > dvcmul | Structured version Visualization version GIF version |
Description: The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
Ref | Expression |
---|---|
dvcmul.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvcmul.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvcmul.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
dvcmul.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
dvcmul.c | ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) |
Ref | Expression |
---|---|
dvcmul | ⊢ (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹))‘𝐶) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcmul.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | fconst6g 6774 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑆 × {𝐴}):𝑆⟶ℂ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑆 × {𝐴}):𝑆⟶ℂ) |
4 | ssidd 4000 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑆) | |
5 | dvcmul.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
6 | dvcmul.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
7 | dvcmul.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
8 | recnprss 25788 | . . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
10 | 9, 5, 6 | dvbss 25785 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋) |
11 | dvcmul.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) | |
12 | 10, 11 | sseldd 3978 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
13 | 6, 12 | sseldd 3978 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
14 | fconst6g 6774 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}):ℂ⟶ℂ) | |
15 | 1, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}):ℂ⟶ℂ) |
16 | ssidd 4000 | . . . . . . . 8 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
17 | dvconst 25801 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) | |
18 | 1, 17 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) |
19 | 18 | dmeqd 5899 | . . . . . . . . . 10 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = dom (ℂ × {0})) |
20 | c0ex 11212 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
21 | 20 | fconst 6771 | . . . . . . . . . . 11 ⊢ (ℂ × {0}):ℂ⟶{0} |
22 | 21 | fdmi 6723 | . . . . . . . . . 10 ⊢ dom (ℂ × {0}) = ℂ |
23 | 19, 22 | eqtrdi 2782 | . . . . . . . . 9 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = ℂ) |
24 | 9, 23 | sseqtrrd 4018 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ dom (ℂ D (ℂ × {𝐴}))) |
25 | dvres3 25797 | . . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ (ℂ × {𝐴}):ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ 𝑆 ⊆ dom (ℂ D (ℂ × {𝐴})))) → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = ((ℂ D (ℂ × {𝐴})) ↾ 𝑆)) | |
26 | 7, 15, 16, 24, 25 | syl22anc 836 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = ((ℂ D (ℂ × {𝐴})) ↾ 𝑆)) |
27 | xpssres 6012 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) | |
28 | 9, 27 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) |
29 | 28 | oveq2d 7421 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = (𝑆 D (𝑆 × {𝐴}))) |
30 | 18 | reseq1d 5974 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = ((ℂ × {0}) ↾ 𝑆)) |
31 | xpssres 6012 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) | |
32 | 9, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) |
33 | 30, 32 | eqtrd 2766 | . . . . . . 7 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = (𝑆 × {0})) |
34 | 26, 29, 33 | 3eqtr3d 2774 | . . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
35 | 20 | fconst2 7202 | . . . . . 6 ⊢ ((𝑆 D (𝑆 × {𝐴})):𝑆⟶{0} ↔ (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
36 | 34, 35 | sylibr 233 | . . . . 5 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})):𝑆⟶{0}) |
37 | 36 | fdmd 6722 | . . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑆 × {𝐴})) = 𝑆) |
38 | 13, 37 | eleqtrrd 2830 | . . 3 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D (𝑆 × {𝐴}))) |
39 | 3, 4, 5, 6, 7, 38, 11 | dvmul 25827 | . 2 ⊢ (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹))‘𝐶) = ((((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) + (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)))) |
40 | 34 | fveq1d 6887 | . . . . . 6 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = ((𝑆 × {0})‘𝐶)) |
41 | 20 | fvconst2 7201 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑆 → ((𝑆 × {0})‘𝐶) = 0) |
42 | 13, 41 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝑆 × {0})‘𝐶) = 0) |
43 | 40, 42 | eqtrd 2766 | . . . . 5 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = 0) |
44 | 43 | oveq1d 7420 | . . . 4 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = (0 · (𝐹‘𝐶))) |
45 | 5, 12 | ffvelcdmd 7081 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
46 | 45 | mul02d 11416 | . . . 4 ⊢ (𝜑 → (0 · (𝐹‘𝐶)) = 0) |
47 | 44, 46 | eqtrd 2766 | . . 3 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = 0) |
48 | fvconst2g 7199 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ 𝑆) → ((𝑆 × {𝐴})‘𝐶) = 𝐴) | |
49 | 1, 13, 48 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ((𝑆 × {𝐴})‘𝐶) = 𝐴) |
50 | 49 | oveq2d 7421 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (((𝑆 D 𝐹)‘𝐶) · 𝐴)) |
51 | dvfg 25790 | . . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
52 | 7, 51 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
53 | 52, 11 | ffvelcdmd 7081 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ) |
54 | 53, 1 | mulcomd 11239 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · 𝐴) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
55 | 50, 54 | eqtrd 2766 | . . 3 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
56 | 47, 55 | oveq12d 7423 | . 2 ⊢ (𝜑 → ((((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) + (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶))) = (0 + (𝐴 · ((𝑆 D 𝐹)‘𝐶)))) |
57 | 1, 53 | mulcld 11238 | . . 3 ⊢ (𝜑 → (𝐴 · ((𝑆 D 𝐹)‘𝐶)) ∈ ℂ) |
58 | 57 | addlidd 11419 | . 2 ⊢ (𝜑 → (0 + (𝐴 · ((𝑆 D 𝐹)‘𝐶))) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
59 | 39, 56, 58 | 3eqtrd 2770 | 1 ⊢ (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹))‘𝐶) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 {csn 4623 {cpr 4625 × cxp 5667 dom cdm 5669 ↾ cres 5671 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∘f cof 7665 ℂcc 11110 ℝcr 11111 0cc0 11112 + caddc 11115 · cmul 11117 D cdv 25747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 |
This theorem is referenced by: (None) |
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