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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 6809 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑆 × {𝐴}):𝑆⟶ℂ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑆 × {𝐴}):𝑆⟶ℂ) |
4 | ssidd 4026 | . . 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 25951 | . . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
10 | 9, 5, 6 | dvbss 25948 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋) |
11 | dvcmul.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) | |
12 | 10, 11 | sseldd 4003 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
13 | 6, 12 | sseldd 4003 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
14 | fconst6g 6809 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}):ℂ⟶ℂ) | |
15 | 1, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}):ℂ⟶ℂ) |
16 | ssidd 4026 | . . . . . . . 8 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
17 | dvconst 25964 | . . . . . . . . . . . 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 5929 | . . . . . . . . . 10 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = dom (ℂ × {0})) |
20 | c0ex 11280 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
21 | 20 | fconst 6806 | . . . . . . . . . . 11 ⊢ (ℂ × {0}):ℂ⟶{0} |
22 | 21 | fdmi 6757 | . . . . . . . . . 10 ⊢ dom (ℂ × {0}) = ℂ |
23 | 19, 22 | eqtrdi 2790 | . . . . . . . . 9 ⊢ (𝜑 → dom (ℂ D (ℂ × {𝐴})) = ℂ) |
24 | 9, 23 | sseqtrrd 4044 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ dom (ℂ D (ℂ × {𝐴}))) |
25 | dvres3 25960 | . . . . . . . 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 838 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = ((ℂ D (ℂ × {𝐴})) ↾ 𝑆)) |
27 | xpssres 6046 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) | |
28 | 9, 27 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) |
29 | 28 | oveq2d 7461 | . . . . . . 7 ⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = (𝑆 D (𝑆 × {𝐴}))) |
30 | 18 | reseq1d 6007 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = ((ℂ × {0}) ↾ 𝑆)) |
31 | xpssres 6046 | . . . . . . . . 9 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) | |
32 | 9, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) |
33 | 30, 32 | eqtrd 2774 | . . . . . . 7 ⊢ (𝜑 → ((ℂ D (ℂ × {𝐴})) ↾ 𝑆) = (𝑆 × {0})) |
34 | 26, 29, 33 | 3eqtr3d 2782 | . . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
35 | 20 | fconst2 7240 | . . . . . 6 ⊢ ((𝑆 D (𝑆 × {𝐴})):𝑆⟶{0} ↔ (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
36 | 34, 35 | sylibr 234 | . . . . 5 ⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})):𝑆⟶{0}) |
37 | 36 | fdmd 6756 | . . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑆 × {𝐴})) = 𝑆) |
38 | 13, 37 | eleqtrrd 2841 | . . 3 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D (𝑆 × {𝐴}))) |
39 | 3, 4, 5, 6, 7, 38, 11 | dvmul 25990 | . 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 6921 | . . . . . 6 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = ((𝑆 × {0})‘𝐶)) |
41 | 20 | fvconst2 7239 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑆 → ((𝑆 × {0})‘𝐶) = 0) |
42 | 13, 41 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝑆 × {0})‘𝐶) = 0) |
43 | 40, 42 | eqtrd 2774 | . . . . 5 ⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴}))‘𝐶) = 0) |
44 | 43 | oveq1d 7460 | . . . 4 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = (0 · (𝐹‘𝐶))) |
45 | 5, 12 | ffvelcdmd 7117 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
46 | 45 | mul02d 11484 | . . . 4 ⊢ (𝜑 → (0 · (𝐹‘𝐶)) = 0) |
47 | 44, 46 | eqtrd 2774 | . . 3 ⊢ (𝜑 → (((𝑆 D (𝑆 × {𝐴}))‘𝐶) · (𝐹‘𝐶)) = 0) |
48 | fvconst2g 7237 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ 𝑆) → ((𝑆 × {𝐴})‘𝐶) = 𝐴) | |
49 | 1, 13, 48 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ((𝑆 × {𝐴})‘𝐶) = 𝐴) |
50 | 49 | oveq2d 7461 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (((𝑆 D 𝐹)‘𝐶) · 𝐴)) |
51 | dvfg 25953 | . . . . . . 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 7117 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ) |
54 | 53, 1 | mulcomd 11307 | . . . 4 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · 𝐴) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
55 | 50, 54 | eqtrd 2774 | . . 3 ⊢ (𝜑 → (((𝑆 D 𝐹)‘𝐶) · ((𝑆 × {𝐴})‘𝐶)) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
56 | 47, 55 | oveq12d 7463 | . 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 11306 | . . 3 ⊢ (𝜑 → (𝐴 · ((𝑆 D 𝐹)‘𝐶)) ∈ ℂ) |
58 | 57 | addlidd 11487 | . 2 ⊢ (𝜑 → (0 + (𝐴 · ((𝑆 D 𝐹)‘𝐶))) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) |
59 | 39, 56, 58 | 3eqtrd 2778 | 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 1537 ∈ wcel 2103 ⊆ wss 3970 {csn 4648 {cpr 4650 × cxp 5697 dom cdm 5699 ↾ cres 5701 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ∘f cof 7708 ℂcc 11178 ℝcr 11179 0cc0 11180 + caddc 11183 · cmul 11185 D cdv 25910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-icc 13410 df-fz 13564 df-fzo 13708 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-mulg 19103 df-cntz 19352 df-cmn 19819 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-lp 23158 df-perf 23159 df-cn 23249 df-cnp 23250 df-haus 23337 df-tx 23584 df-hmeo 23777 df-fil 23868 df-fm 23960 df-flim 23961 df-flf 23962 df-xms 24344 df-ms 24345 df-tms 24346 df-cncf 24916 df-limc 25913 df-dv 25914 |
This theorem is referenced by: (None) |
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