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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvsubf | Structured version Visualization version GIF version |
Description: The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvsubf.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvsubf.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvsubf.g | ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
dvsubf.fdv | ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
dvsubf.gdv | ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) |
Ref | Expression |
---|---|
dvsubf | ⊢ (𝜑 → (𝑆 D (𝐹 ∘f − 𝐺)) = ((𝑆 D 𝐹) ∘f − (𝑆 D 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvsubf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvsubf.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
3 | 2 | ffvelrnda 6828 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℂ) |
4 | dvfg 24509 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
6 | dvsubf.fdv | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) | |
7 | 6 | feq2d 6473 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
8 | 5, 7 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
9 | 8 | ffvelrnda 6828 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) ∈ ℂ) |
10 | 2 | feqmptd 6708 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥))) |
11 | 10 | oveq2d 7151 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥)))) |
12 | 8 | feqmptd 6708 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑥))) |
13 | 11, 12 | eqtr3d 2835 | . . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑥))) |
14 | dvsubf.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) | |
15 | 14 | ffvelrnda 6828 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ℂ) |
16 | dvfg 24509 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) | |
17 | 1, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) |
18 | dvsubf.gdv | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) | |
19 | 18 | feq2d 6473 | . . . . 5 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ)) |
20 | 17, 19 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ) |
21 | 20 | ffvelrnda 6828 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐺)‘𝑥) ∈ ℂ) |
22 | 14 | feqmptd 6708 | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ (𝐺‘𝑥))) |
23 | 22 | oveq2d 7151 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐺‘𝑥)))) |
24 | 20 | feqmptd 6708 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D 𝐺)‘𝑥))) |
25 | 23, 24 | eqtr3d 2835 | . . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐺‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D 𝐺)‘𝑥))) |
26 | 1, 3, 9, 13, 15, 21, 25 | dvmptsub 24570 | . 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) − (𝐺‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D 𝐹)‘𝑥) − ((𝑆 D 𝐺)‘𝑥)))) |
27 | ovex 7168 | . . . . . 6 ⊢ (𝑆 D 𝐹) ∈ V | |
28 | 27 | dmex 7598 | . . . . 5 ⊢ dom (𝑆 D 𝐹) ∈ V |
29 | 6, 28 | eqeltrrdi 2899 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) |
30 | 29, 3, 15, 10, 22 | offval2 7406 | . . 3 ⊢ (𝜑 → (𝐹 ∘f − 𝐺) = (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) − (𝐺‘𝑥)))) |
31 | 30 | oveq2d 7151 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f − 𝐺)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) − (𝐺‘𝑥))))) |
32 | 29, 9, 21, 12, 24 | offval2 7406 | . 2 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘f − (𝑆 D 𝐺)) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D 𝐹)‘𝑥) − ((𝑆 D 𝐺)‘𝑥)))) |
33 | 26, 31, 32 | 3eqtr4d 2843 | 1 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f − 𝐺)) = ((𝑆 D 𝐹) ∘f − (𝑆 D 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {cpr 4527 ↦ cmpt 5110 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 ℂcc 10524 ℝcr 10525 − cmin 10859 D cdv 24466 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 |
This theorem is referenced by: dvsubcncf 42566 |
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