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Mirrors > Home > ILE Home > Th. List > dvmptaddx | GIF version |
Description: Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvmptadd.s | β’ (π β π β {β, β}) |
dvmptadd.a | β’ ((π β§ π₯ β π) β π΄ β β) |
dvmptadd.b | β’ ((π β§ π₯ β π) β π΅ β π) |
dvmptadd.da | β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) |
dvmptclx.ss | β’ (π β π β π) |
dvmptadd.c | β’ ((π β§ π₯ β π) β πΆ β β) |
dvmptadd.d | β’ ((π β§ π₯ β π) β π· β π) |
dvmptadd.dc | β’ (π β (π D (π₯ β π β¦ πΆ)) = (π₯ β π β¦ π·)) |
Ref | Expression |
---|---|
dvmptaddx | β’ (π β (π D (π₯ β π β¦ (π΄ + πΆ))) = (π₯ β π β¦ (π΅ + π·))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptadd.s | . . 3 β’ (π β π β {β, β}) | |
2 | dvmptclx.ss | . . 3 β’ (π β π β π) | |
3 | dvmptadd.a | . . . 4 β’ ((π β§ π₯ β π) β π΄ β β) | |
4 | 3 | fmpttd 5673 | . . 3 β’ (π β (π₯ β π β¦ π΄):πβΆβ) |
5 | dvmptadd.c | . . . 4 β’ ((π β§ π₯ β π) β πΆ β β) | |
6 | 5 | fmpttd 5673 | . . 3 β’ (π β (π₯ β π β¦ πΆ):πβΆβ) |
7 | dvmptadd.da | . . . . 5 β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) | |
8 | 7 | dmeqd 4831 | . . . 4 β’ (π β dom (π D (π₯ β π β¦ π΄)) = dom (π₯ β π β¦ π΅)) |
9 | dvmptadd.b | . . . . . 6 β’ ((π β§ π₯ β π) β π΅ β π) | |
10 | 9 | ralrimiva 2550 | . . . . 5 β’ (π β βπ₯ β π π΅ β π) |
11 | dmmptg 5128 | . . . . 5 β’ (βπ₯ β π π΅ β π β dom (π₯ β π β¦ π΅) = π) | |
12 | 10, 11 | syl 14 | . . . 4 β’ (π β dom (π₯ β π β¦ π΅) = π) |
13 | 8, 12 | eqtrd 2210 | . . 3 β’ (π β dom (π D (π₯ β π β¦ π΄)) = π) |
14 | dvmptadd.dc | . . . . 5 β’ (π β (π D (π₯ β π β¦ πΆ)) = (π₯ β π β¦ π·)) | |
15 | 14 | dmeqd 4831 | . . . 4 β’ (π β dom (π D (π₯ β π β¦ πΆ)) = dom (π₯ β π β¦ π·)) |
16 | dvmptadd.d | . . . . . 6 β’ ((π β§ π₯ β π) β π· β π) | |
17 | 16 | ralrimiva 2550 | . . . . 5 β’ (π β βπ₯ β π π· β π) |
18 | dmmptg 5128 | . . . . 5 β’ (βπ₯ β π π· β π β dom (π₯ β π β¦ π·) = π) | |
19 | 17, 18 | syl 14 | . . . 4 β’ (π β dom (π₯ β π β¦ π·) = π) |
20 | 15, 19 | eqtrd 2210 | . . 3 β’ (π β dom (π D (π₯ β π β¦ πΆ)) = π) |
21 | 1, 2, 4, 6, 13, 20 | dviaddf 14254 | . 2 β’ (π β (π D ((π₯ β π β¦ π΄) βπ + (π₯ β π β¦ πΆ))) = ((π D (π₯ β π β¦ π΄)) βπ + (π D (π₯ β π β¦ πΆ)))) |
22 | 1, 2 | ssexd 4145 | . . . 4 β’ (π β π β V) |
23 | eqidd 2178 | . . . 4 β’ (π β (π₯ β π β¦ π΄) = (π₯ β π β¦ π΄)) | |
24 | eqidd 2178 | . . . 4 β’ (π β (π₯ β π β¦ πΆ) = (π₯ β π β¦ πΆ)) | |
25 | 22, 3, 5, 23, 24 | offval2 6100 | . . 3 β’ (π β ((π₯ β π β¦ π΄) βπ + (π₯ β π β¦ πΆ)) = (π₯ β π β¦ (π΄ + πΆ))) |
26 | 25 | oveq2d 5893 | . 2 β’ (π β (π D ((π₯ β π β¦ π΄) βπ + (π₯ β π β¦ πΆ))) = (π D (π₯ β π β¦ (π΄ + πΆ)))) |
27 | 22, 9, 16, 7, 14 | offval2 6100 | . 2 β’ (π β ((π D (π₯ β π β¦ π΄)) βπ + (π D (π₯ β π β¦ πΆ))) = (π₯ β π β¦ (π΅ + π·))) |
28 | 21, 26, 27 | 3eqtr3d 2218 | 1 β’ (π β (π D (π₯ β π β¦ (π΄ + πΆ))) = (π₯ β π β¦ (π΅ + π·))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βwral 2455 Vcvv 2739 β wss 3131 {cpr 3595 β¦ cmpt 4066 dom cdm 4628 (class class class)co 5877 βπ cof 6083 βcc 7811 βcr 7812 + caddc 7816 D cdv 14209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 ax-addf 7935 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-of 6085 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-map 6652 df-pm 6653 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-rest 12695 df-topgen 12714 df-psmet 13532 df-xmet 13533 df-met 13534 df-bl 13535 df-mopn 13536 df-top 13583 df-topon 13596 df-bases 13628 df-ntr 13681 df-cn 13773 df-cnp 13774 df-tx 13838 df-limced 14210 df-dvap 14211 |
This theorem is referenced by: dvmptsubcn 14270 |
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