![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dvconst | Structured version Visualization version GIF version |
Description: Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
dvconst | β’ (π΄ β β β (β D (β Γ {π΄})) = (β Γ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst6g 6773 | . 2 β’ (π΄ β β β (β Γ {π΄}):ββΆβ) | |
2 | simpr2 1192 | . . . . . . 7 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β π§ β β) | |
3 | fvconst2g 7198 | . . . . . . 7 β’ ((π΄ β β β§ π§ β β) β ((β Γ {π΄})βπ§) = π΄) | |
4 | 2, 3 | syldan 590 | . . . . . 6 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β ((β Γ {π΄})βπ§) = π΄) |
5 | fvconst2g 7198 | . . . . . . 7 β’ ((π΄ β β β§ π₯ β β) β ((β Γ {π΄})βπ₯) = π΄) | |
6 | 5 | 3ad2antr1 1185 | . . . . . 6 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β ((β Γ {π΄})βπ₯) = π΄) |
7 | 4, 6 | oveq12d 7422 | . . . . 5 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β (((β Γ {π΄})βπ§) β ((β Γ {π΄})βπ₯)) = (π΄ β π΄)) |
8 | subid 11480 | . . . . . 6 β’ (π΄ β β β (π΄ β π΄) = 0) | |
9 | 8 | adantr 480 | . . . . 5 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β (π΄ β π΄) = 0) |
10 | 7, 9 | eqtrd 2766 | . . . 4 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β (((β Γ {π΄})βπ§) β ((β Γ {π΄})βπ₯)) = 0) |
11 | 10 | oveq1d 7419 | . . 3 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β ((((β Γ {π΄})βπ§) β ((β Γ {π΄})βπ₯)) / (π§ β π₯)) = (0 / (π§ β π₯))) |
12 | simpr1 1191 | . . . . 5 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β π₯ β β) | |
13 | 2, 12 | subcld 11572 | . . . 4 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β (π§ β π₯) β β) |
14 | simpr3 1193 | . . . . 5 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β π§ β π₯) | |
15 | 2, 12, 14 | subne0d 11581 | . . . 4 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β (π§ β π₯) β 0) |
16 | 13, 15 | div0d 11990 | . . 3 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β (0 / (π§ β π₯)) = 0) |
17 | 11, 16 | eqtrd 2766 | . 2 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β ((((β Γ {π΄})βπ§) β ((β Γ {π΄})βπ₯)) / (π§ β π₯)) = 0) |
18 | 0cn 11207 | . 2 β’ 0 β β | |
19 | 1, 17, 18 | dvidlem 25795 | 1 β’ (π΄ β β β (β D (β Γ {π΄})) = (β Γ {0})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 {csn 4623 Γ cxp 5667 βcfv 6536 (class class class)co 7404 βcc 11107 0cc0 11109 β cmin 11445 / cdiv 11872 D cdv 25743 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-icc 13334 df-fz 13488 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-topgen 17396 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-cncf 24749 df-limc 25746 df-dv 25747 |
This theorem is referenced by: dvcmul 25826 dvcmulf 25827 dvexp2 25837 dvmptc 25841 dvef 25863 dvsconst 43646 binomcxplemnotnn0 43672 |
Copyright terms: Public domain | W3C validator |