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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 6307 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑆 × {𝐴}):𝑆⟶ℂ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑆 × {𝐴}):𝑆⟶ℂ) |
4 | ssidd 3818 | . . 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 24006 | . . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
10 | 9, 5, 6 | dvbss 24003 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋) |
11 | dvcmul.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) | |
12 | 10, 11 | sseldd 3797 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
13 | 6, 12 | sseldd 3797 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
14 | fconst6g 6307 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}):ℂ⟶ℂ) | |
15 | 1, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}):ℂ⟶ℂ) |
16 | ssidd 3818 | . . . . . . . 8 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
17 | dvconst 24018 | . . . . . . . . . . . 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 5527 | . . . . . . . . . 10 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = dom (ℂ × {0})) |
20 | c0ex 10320 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
21 | 20 | fconst 6304 | . . . . . . . . . . 11 ⊢ (ℂ × {0}):ℂ⟶{0} |
22 | 21 | fdmi 6264 | . . . . . . . . . 10 ⊢ dom (ℂ × {0}) = ℂ |
23 | 19, 22 | syl6eq 2847 | . . . . . . . . 9 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = ℂ) |
24 | 9, 23 | sseqtr4d 3836 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ dom (ℂ D (ℂ × {𝐴}))) |
25 | dvres3 24015 | . . . . . . . 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 868 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = ((ℂ D (ℂ × {𝐴})) ↾ 𝑆)) |
27 | xpssres 5641 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) | |
28 | 9, 27 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) |
29 | 28 | oveq2d 6892 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = (𝑆 D (𝑆 × {𝐴}))) |
30 | 18 | reseq1d 5597 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = ((ℂ × {0}) ↾ 𝑆)) |
31 | xpssres 5641 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) | |
32 | 9, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) |
33 | 30, 32 | eqtrd 2831 | . . . . . . 7 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = (𝑆 × {0})) |
34 | 26, 29, 33 | 3eqtr3d 2839 | . . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
35 | 20 | fconst2 6697 | . . . . . 6 ⊢ ((𝑆 D (𝑆 × {𝐴})):𝑆⟶{0} ↔ (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
36 | 34, 35 | sylibr 226 | . . . . 5 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})):𝑆⟶{0}) |
37 | 36 | fdmd 6263 | . . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑆 × {𝐴})) = 𝑆) |
38 | 13, 37 | eleqtrrd 2879 | . . 3 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D (𝑆 × {𝐴}))) |
39 | 3, 4, 5, 6, 7, 38, 11 | dvmul 24042 | . 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 6411 | . . . . . 6 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = ((𝑆 × {0})‘𝐶)) |
41 | 20 | fvconst2 6696 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑆 → ((𝑆 × {0})‘𝐶) = 0) |
42 | 13, 41 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝑆 × {0})‘𝐶) = 0) |
43 | 40, 42 | eqtrd 2831 | . . . . 5 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = 0) |
44 | 43 | oveq1d 6891 | . . . 4 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = (0 · (𝐹‘𝐶))) |
45 | 5, 12 | ffvelrnd 6584 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
46 | 45 | mul02d 10522 | . . . 4 ⊢ (𝜑 → (0 · (𝐹‘𝐶)) = 0) |
47 | 44, 46 | eqtrd 2831 | . . 3 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = 0) |
48 | fvconst2g 6694 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ 𝑆) → ((𝑆 × {𝐴})‘𝐶) = 𝐴) | |
49 | 1, 13, 48 | syl2anc 580 | . . . . 5 ⊢ (𝜑 → ((𝑆 × {𝐴})‘𝐶) = 𝐴) |
50 | 49 | oveq2d 6892 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (((𝑆 D 𝐹)‘𝐶) · 𝐴)) |
51 | dvfg 24008 | . . . . . . 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 | ffvelrnd 6584 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ) |
54 | 53, 1 | mulcomd 10348 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · 𝐴) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
55 | 50, 54 | eqtrd 2831 | . . 3 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
56 | 47, 55 | oveq12d 6894 | . 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 10347 | . . 3 ⊢ (𝜑 → (𝐴 · ((𝑆 D 𝐹)‘𝐶)) ∈ ℂ) |
58 | 57 | addid2d 10525 | . 2 ⊢ (𝜑 → (0 + (𝐴 · ((𝑆 D 𝐹)‘𝐶))) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
59 | 39, 56, 58 | 3eqtrd 2835 | 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 1653 ∈ wcel 2157 ⊆ wss 3767 {csn 4366 {cpr 4368 × cxp 5308 dom cdm 5310 ↾ cres 5312 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 ∘𝑓 cof 7127 ℂcc 10220 ℝcr 10221 0cc0 10222 + caddc 10225 · cmul 10227 D cdv 23965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-fi 8557 df-sup 8588 df-inf 8589 df-oi 8655 df-card 9049 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-q 12030 df-rp 12071 df-xneg 12189 df-xadd 12190 df-xmul 12191 df-icc 12427 df-fz 12577 df-fzo 12717 df-seq 13052 df-exp 13111 df-hash 13367 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-rest 16395 df-topn 16396 df-0g 16414 df-gsum 16415 df-topgen 16416 df-pt 16417 df-prds 16420 df-xrs 16474 df-qtop 16479 df-imas 16480 df-xps 16482 df-mre 16558 df-mrc 16559 df-acs 16561 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-mulg 17854 df-cntz 18059 df-cmn 18507 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-fbas 20062 df-fg 20063 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cld 21149 df-ntr 21150 df-cls 21151 df-nei 21228 df-lp 21266 df-perf 21267 df-cn 21357 df-cnp 21358 df-haus 21445 df-tx 21691 df-hmeo 21884 df-fil 21975 df-fm 22067 df-flim 22068 df-flf 22069 df-xms 22450 df-ms 22451 df-tms 22452 df-cncf 23006 df-limc 23968 df-dv 23969 |
This theorem is referenced by: (None) |
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