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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 6094 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑆 × {𝐴}):𝑆⟶ℂ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑆 × {𝐴}):𝑆⟶ℂ) |
4 | ssid 3624 | . . . 4 ⊢ 𝑆 ⊆ 𝑆 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑆) |
6 | dvcmul.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
7 | dvcmul.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
8 | dvcmul.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
9 | recnprss 23668 | . . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
11 | 10, 6, 7 | dvbss 23665 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋) |
12 | dvcmul.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) | |
13 | 11, 12 | sseldd 3604 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
14 | 7, 13 | sseldd 3604 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
15 | fconst6g 6094 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}):ℂ⟶ℂ) | |
16 | 1, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}):ℂ⟶ℂ) |
17 | ssid 3624 | . . . . . . . . 9 ⊢ ℂ ⊆ ℂ | |
18 | 17 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℂ ⊆ ℂ) |
19 | dvconst 23680 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) | |
20 | 1, 19 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) |
21 | 20 | dmeqd 5326 | . . . . . . . . . 10 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = dom (ℂ × {0})) |
22 | c0ex 10034 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
23 | 22 | fconst 6091 | . . . . . . . . . . 11 ⊢ (ℂ × {0}):ℂ⟶{0} |
24 | 23 | fdmi 6052 | . . . . . . . . . 10 ⊢ dom (ℂ × {0}) = ℂ |
25 | 21, 24 | syl6eq 2672 | . . . . . . . . 9 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = ℂ) |
26 | 10, 25 | sseqtr4d 3642 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ dom (ℂ D (ℂ × {𝐴}))) |
27 | dvres3 23677 | . . . . . . . 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)) | |
28 | 8, 16, 18, 26, 27 | syl22anc 1327 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = ((ℂ D (ℂ × {𝐴})) ↾ 𝑆)) |
29 | xpssres 5434 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) | |
30 | 10, 29 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) |
31 | 30 | oveq2d 6666 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = (𝑆 D (𝑆 × {𝐴}))) |
32 | 20 | reseq1d 5395 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = ((ℂ × {0}) ↾ 𝑆)) |
33 | xpssres 5434 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) | |
34 | 10, 33 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) |
35 | 32, 34 | eqtrd 2656 | . . . . . . 7 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = (𝑆 × {0})) |
36 | 28, 31, 35 | 3eqtr3d 2664 | . . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
37 | 22 | fconst2 6470 | . . . . . 6 ⊢ ((𝑆 D (𝑆 × {𝐴})):𝑆⟶{0} ↔ (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
38 | 36, 37 | sylibr 224 | . . . . 5 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})):𝑆⟶{0}) |
39 | fdm 6051 | . . . . 5 ⊢ ((𝑆 D (𝑆 × {𝐴})):𝑆⟶{0} → dom (𝑆 D (𝑆 × {𝐴})) = 𝑆) | |
40 | 38, 39 | syl 17 | . . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑆 × {𝐴})) = 𝑆) |
41 | 14, 40 | eleqtrrd 2704 | . . 3 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D (𝑆 × {𝐴}))) |
42 | 3, 5, 6, 7, 8, 41, 12 | dvmul 23704 | . 2 ⊢ (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘𝑓 · 𝐹))‘𝐶) = ((((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) + (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)))) |
43 | 36 | fveq1d 6193 | . . . . . 6 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = ((𝑆 × {0})‘𝐶)) |
44 | 22 | fvconst2 6469 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑆 → ((𝑆 × {0})‘𝐶) = 0) |
45 | 14, 44 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝑆 × {0})‘𝐶) = 0) |
46 | 43, 45 | eqtrd 2656 | . . . . 5 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = 0) |
47 | 46 | oveq1d 6665 | . . . 4 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = (0 · (𝐹‘𝐶))) |
48 | 6, 13 | ffvelrnd 6360 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
49 | 48 | mul02d 10234 | . . . 4 ⊢ (𝜑 → (0 · (𝐹‘𝐶)) = 0) |
50 | 47, 49 | eqtrd 2656 | . . 3 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = 0) |
51 | fvconst2g 6467 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ 𝑆) → ((𝑆 × {𝐴})‘𝐶) = 𝐴) | |
52 | 1, 14, 51 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → ((𝑆 × {𝐴})‘𝐶) = 𝐴) |
53 | 52 | oveq2d 6666 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (((𝑆 D 𝐹)‘𝐶) · 𝐴)) |
54 | dvfg 23670 | . . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
55 | 8, 54 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
56 | 55, 12 | ffvelrnd 6360 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ) |
57 | 56, 1 | mulcomd 10061 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · 𝐴) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
58 | 53, 57 | eqtrd 2656 | . . 3 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
59 | 50, 58 | oveq12d 6668 | . 2 ⊢ (𝜑 → ((((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) + (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶))) = (0 + (𝐴 · ((𝑆 D 𝐹)‘𝐶)))) |
60 | 1, 56 | mulcld 10060 | . . 3 ⊢ (𝜑 → (𝐴 · ((𝑆 D 𝐹)‘𝐶)) ∈ ℂ) |
61 | 60 | addid2d 10237 | . 2 ⊢ (𝜑 → (0 + (𝐴 · ((𝑆 D 𝐹)‘𝐶))) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
62 | 42, 59, 61 | 3eqtrd 2660 | 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 1483 ∈ wcel 1990 ⊆ wss 3574 {csn 4177 {cpr 4179 × cxp 5112 dom cdm 5114 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 · cmul 9941 D cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 |
This theorem is referenced by: (None) |
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