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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 6780 | . 2 β’ (π΄ β β β (β Γ {π΄}):ββΆβ) | |
2 | simpr2 1195 | . . . . . . 7 β’ ((π΄ β β β§ (π₯ β β β§ π§ β β β§ π§ β π₯)) β π§ β β) | |
3 | fvconst2g 7202 | . . . . . . 7 β’ ((π΄ β β β§ π§ β β) β ((β Γ {π΄})βπ§) = π΄) | |
4 | 2, 3 | syldan 591 | . . . . . 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 7202 | . . . . . . 7 β’ ((π΄ β β β§ π₯ β β) β ((β Γ {π΄})βπ₯) = π΄) | |
6 | 5 | 3ad2antr1 1188 | . . . . . 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 7426 | . . . . 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 11478 | . . . . . 6 β’ (π΄ β β β (π΄ β π΄) = 0) | |
9 | 8 | adantr 481 | . . . . 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 2772 | . . . 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 7423 | . . 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 1194 | . . . . 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 11570 | . . . 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 1196 | . . . . 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 11579 | . . . 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 11988 | . . 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 2772 | . 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 11205 | . 2 β’ 0 β β | |
19 | 1, 17, 18 | dvidlem 25431 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 {csn 4628 Γ cxp 5674 βcfv 6543 (class class class)co 7408 βcc 11107 0cc0 11109 β cmin 11443 / cdiv 11870 D cdv 25379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 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 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-icc 13330 df-fz 13484 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17367 df-topn 17368 df-topgen 17388 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-cncf 24393 df-limc 25382 df-dv 25383 |
This theorem is referenced by: dvcmul 25460 dvcmulf 25461 dvexp2 25470 dvmptc 25474 dvef 25496 dvsconst 43079 binomcxplemnotnn0 43105 |
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