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Mirrors > Home > MPE Home > Th. List > dvaddf | Structured version Visualization version GIF version |
Description: The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
Ref | Expression |
---|---|
dvaddf.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvaddf.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvaddf.g | ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
dvaddf.df | ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
dvaddf.dg | ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) |
Ref | Expression |
---|---|
dvaddf | ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)) = ((𝑆 D 𝐹) ∘f + (𝑆 D 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvaddf.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvaddf.df | . . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) | |
3 | dvbsss 24500 | . . . . 5 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | |
4 | 2, 3 | eqsstrrdi 4022 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
5 | 1, 4 | ssexd 5228 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
6 | dvfg 24504 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
8 | 2 | feq2d 6500 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
9 | 7, 8 | mpbid 234 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
10 | 9 | ffnd 6515 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐹) Fn 𝑋) |
11 | dvfg 24504 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) | |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) |
13 | dvaddf.dg | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) | |
14 | 13 | feq2d 6500 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ)) |
15 | 12, 14 | mpbid 234 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ) |
16 | 15 | ffnd 6515 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋) |
17 | dvfg 24504 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (𝐹 ∘f + 𝐺)):dom (𝑆 D (𝐹 ∘f + 𝐺))⟶ℂ) | |
18 | 1, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)):dom (𝑆 D (𝐹 ∘f + 𝐺))⟶ℂ) |
19 | recnprss 24502 | . . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
20 | 1, 19 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
21 | addcl 10619 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
22 | 21 | adantl 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ) |
23 | dvaddf.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
24 | dvaddf.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) | |
25 | inidm 4195 | . . . . . . . . 9 ⊢ (𝑋 ∩ 𝑋) = 𝑋 | |
26 | 22, 23, 24, 5, 5, 25 | off 7424 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝑋⟶ℂ) |
27 | 20, 26, 4 | dvbss 24499 | . . . . . . 7 ⊢ (𝜑 → dom (𝑆 D (𝐹 ∘f + 𝐺)) ⊆ 𝑋) |
28 | 23 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶ℂ) |
29 | 4 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑋 ⊆ 𝑆) |
30 | 24 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺:𝑋⟶ℂ) |
31 | 20 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ ℂ) |
32 | fvexd 6685 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) ∈ V) | |
33 | fvexd 6685 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐺)‘𝑥) ∈ V) | |
34 | 2 | eleq2d 2898 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥 ∈ 𝑋)) |
35 | 34 | biimpar 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D 𝐹)) |
36 | 1 | adantr 483 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ {ℝ, ℂ}) |
37 | ffun 6517 | . . . . . . . . . . 11 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
38 | funfvbrb 6821 | . . . . . . . . . . 11 ⊢ (Fun (𝑆 D 𝐹) → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥))) | |
39 | 36, 6, 37, 38 | 4syl 19 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥))) |
40 | 35, 39 | mpbid 234 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥)) |
41 | 13 | eleq2d 2898 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥 ∈ 𝑋)) |
42 | 41 | biimpar 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D 𝐺)) |
43 | ffun 6517 | . . . . . . . . . . 11 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺)) | |
44 | funfvbrb 6821 | . . . . . . . . . . 11 ⊢ (Fun (𝑆 D 𝐺) → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥))) | |
45 | 36, 11, 43, 44 | 4syl 19 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥))) |
46 | 42, 45 | mpbid 234 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥)) |
47 | eqid 2821 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
48 | 28, 29, 30, 29, 31, 32, 33, 40, 46, 47 | dvaddbr 24535 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D (𝐹 ∘f + 𝐺))(((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥))) |
49 | reldv 24468 | . . . . . . . . 9 ⊢ Rel (𝑆 D (𝐹 ∘f + 𝐺)) | |
50 | 49 | releldmi 5818 | . . . . . . . 8 ⊢ (𝑥(𝑆 D (𝐹 ∘f + 𝐺))(((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥)) → 𝑥 ∈ dom (𝑆 D (𝐹 ∘f + 𝐺))) |
51 | 48, 50 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D (𝐹 ∘f + 𝐺))) |
52 | 27, 51 | eqelssd 3988 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D (𝐹 ∘f + 𝐺)) = 𝑋) |
53 | 52 | feq2d 6500 | . . . . 5 ⊢ (𝜑 → ((𝑆 D (𝐹 ∘f + 𝐺)):dom (𝑆 D (𝐹 ∘f + 𝐺))⟶ℂ ↔ (𝑆 D (𝐹 ∘f + 𝐺)):𝑋⟶ℂ)) |
54 | 18, 53 | mpbid 234 | . . . 4 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)):𝑋⟶ℂ) |
55 | 54 | ffnd 6515 | . . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)) Fn 𝑋) |
56 | eqidd 2822 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) = ((𝑆 D 𝐹)‘𝑥)) | |
57 | eqidd 2822 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐺)‘𝑥) = ((𝑆 D 𝐺)‘𝑥)) | |
58 | 28, 29, 30, 29, 36, 35, 42 | dvadd 24537 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D (𝐹 ∘f + 𝐺))‘𝑥) = (((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥))) |
59 | 58 | eqcomd 2827 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥)) = ((𝑆 D (𝐹 ∘f + 𝐺))‘𝑥)) |
60 | 5, 10, 16, 55, 56, 57, 59 | offveq 7430 | . 2 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘f + (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∘f + 𝐺))) |
61 | 60 | eqcomd 2827 | 1 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)) = ((𝑆 D 𝐹) ∘f + (𝑆 D 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {cpr 4569 class class class wbr 5066 dom cdm 5555 Fun wfun 6349 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 ℂcc 10535 ℝcr 10536 + caddc 10540 TopOpenctopn 16695 ℂfldccnfld 20545 D cdv 24461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-limc 24464 df-dv 24465 |
This theorem is referenced by: dvmptadd 24557 |
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