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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 25952 | . . . . 5 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | |
4 | 2, 3 | eqsstrrdi 4051 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
5 | 1, 4 | ssexd 5330 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
6 | dvfg 25956 | . . . . . 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 6723 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
9 | 7, 8 | mpbid 232 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
10 | 9 | ffnd 6738 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐹) Fn 𝑋) |
11 | dvfg 25956 | . . . . . 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 6723 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ)) |
15 | 12, 14 | mpbid 232 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ) |
16 | 15 | ffnd 6738 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋) |
17 | dvfg 25956 | . . . . . 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 25954 | . . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
20 | 1, 19 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
21 | addcl 11235 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
22 | 21 | adantl 481 | . . . . . . . . 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 4235 | . . . . . . . . 9 ⊢ (𝑋 ∩ 𝑋) = 𝑋 | |
26 | 22, 23, 24, 5, 5, 25 | off 7715 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝑋⟶ℂ) |
27 | 20, 26, 4 | dvbss 25951 | . . . . . . 7 ⊢ (𝜑 → dom (𝑆 D (𝐹 ∘f + 𝐺)) ⊆ 𝑋) |
28 | 23 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶ℂ) |
29 | 4 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑋 ⊆ 𝑆) |
30 | 24 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺:𝑋⟶ℂ) |
31 | 20 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ ℂ) |
32 | 2 | eleq2d 2825 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥 ∈ 𝑋)) |
33 | 32 | biimpar 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D 𝐹)) |
34 | 1 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ {ℝ, ℂ}) |
35 | ffun 6740 | . . . . . . . . . . 11 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
36 | funfvbrb 7071 | . . . . . . . . . . 11 ⊢ (Fun (𝑆 D 𝐹) → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥))) | |
37 | 34, 6, 35, 36 | 4syl 19 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥))) |
38 | 33, 37 | mpbid 232 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥)) |
39 | 13 | eleq2d 2825 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥 ∈ 𝑋)) |
40 | 39 | biimpar 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D 𝐺)) |
41 | ffun 6740 | . . . . . . . . . . 11 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺)) | |
42 | funfvbrb 7071 | . . . . . . . . . . 11 ⊢ (Fun (𝑆 D 𝐺) → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥))) | |
43 | 34, 11, 41, 42 | 4syl 19 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥))) |
44 | 40, 43 | mpbid 232 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥)) |
45 | eqid 2735 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
46 | 28, 29, 30, 29, 31, 38, 44, 45 | dvaddbr 25989 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D (𝐹 ∘f + 𝐺))(((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥))) |
47 | reldv 25920 | . . . . . . . . 9 ⊢ Rel (𝑆 D (𝐹 ∘f + 𝐺)) | |
48 | 47 | releldmi 5962 | . . . . . . . 8 ⊢ (𝑥(𝑆 D (𝐹 ∘f + 𝐺))(((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥)) → 𝑥 ∈ dom (𝑆 D (𝐹 ∘f + 𝐺))) |
49 | 46, 48 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D (𝐹 ∘f + 𝐺))) |
50 | 27, 49 | eqelssd 4017 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D (𝐹 ∘f + 𝐺)) = 𝑋) |
51 | 50 | feq2d 6723 | . . . . 5 ⊢ (𝜑 → ((𝑆 D (𝐹 ∘f + 𝐺)):dom (𝑆 D (𝐹 ∘f + 𝐺))⟶ℂ ↔ (𝑆 D (𝐹 ∘f + 𝐺)):𝑋⟶ℂ)) |
52 | 18, 51 | mpbid 232 | . . . 4 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)):𝑋⟶ℂ) |
53 | 52 | ffnd 6738 | . . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)) Fn 𝑋) |
54 | eqidd 2736 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) = ((𝑆 D 𝐹)‘𝑥)) | |
55 | eqidd 2736 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐺)‘𝑥) = ((𝑆 D 𝐺)‘𝑥)) | |
56 | 28, 29, 30, 29, 34, 33, 40 | dvadd 25992 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D (𝐹 ∘f + 𝐺))‘𝑥) = (((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥))) |
57 | 56 | eqcomd 2741 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D 𝐹)‘𝑥) + ((𝑆 D 𝐺)‘𝑥)) = ((𝑆 D (𝐹 ∘f + 𝐺))‘𝑥)) |
58 | 5, 10, 16, 53, 54, 55, 57 | offveq 7723 | . 2 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘f + (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∘f + 𝐺))) |
59 | 58 | eqcomd 2741 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 {cpr 4633 class class class wbr 5148 dom cdm 5689 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ℂcc 11151 ℝcr 11152 + caddc 11156 TopOpenctopn 17468 ℂfldccnfld 21382 D cdv 25913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 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 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-limc 25916 df-dv 25917 |
This theorem is referenced by: dvmptadd 26013 |
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