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Mirrors > Home > MPE Home > Th. List > dvn2bss | Structured version Visualization version GIF version |
Description: An N-times differentiable point is an M-times differentiable point, if 𝑀 ≤ 𝑁. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
dvn2bss | ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ}) | |
2 | simp2 1134 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
3 | elfznn0 12995 | . . . . . 6 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0) | |
4 | 3 | 3ad2ant3 1132 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ ℕ0) |
5 | elfzuz3 12899 | . . . . . . 7 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
6 | 5 | 3ad2ant3 1132 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑀)) |
7 | uznn0sub 12265 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝑁 − 𝑀) ∈ ℕ0) |
9 | dvnadd 24532 | . . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑀 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ0)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀)))) | |
10 | 1, 2, 4, 8, 9 | syl22anc 837 | . . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀)))) |
11 | 4 | nn0cnd 11945 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ ℂ) |
12 | elfzuz2 12907 | . . . . . . . . 9 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘0)) | |
13 | 12 | 3ad2ant3 1132 | . . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ≥‘0)) |
14 | nn0uz 12268 | . . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0) | |
15 | 13, 14 | eleqtrrdi 2901 | . . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0) |
16 | 15 | nn0cnd 11945 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℂ) |
17 | 11, 16 | pncan3d 10989 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝑀 + (𝑁 − 𝑀)) = 𝑁) |
18 | 17 | fveq2d 6649 | . . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
19 | 10, 18 | eqtrd 2833 | . . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
20 | 19 | dmeqd 5738 | . 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
21 | cnex 10607 | . . . . 5 ⊢ ℂ ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ℂ ∈ V) |
23 | dvnf 24530 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ) | |
24 | 3, 23 | syl3an3 1162 | . . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ) |
25 | dvnbss 24531 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) | |
26 | 3, 25 | syl3an3 1162 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) |
27 | elpmi 8408 | . . . . . . 7 ⊢ (𝐹 ∈ (ℂ ↑pm 𝑆) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)) | |
28 | 27 | 3ad2ant2 1131 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)) |
29 | 28 | simprd 499 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom 𝐹 ⊆ 𝑆) |
30 | 26, 29 | sstrd 3925 | . . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝑆) |
31 | elpm2r 8407 | . . . 4 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ ∧ dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝑆)) → ((𝑆 D𝑛 𝐹)‘𝑀) ∈ (ℂ ↑pm 𝑆)) | |
32 | 22, 1, 24, 30, 31 | syl22anc 837 | . . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑀) ∈ (ℂ ↑pm 𝑆)) |
33 | dvnbss 24531 | . . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ ((𝑆 D𝑛 𝐹)‘𝑀) ∈ (ℂ ↑pm 𝑆) ∧ (𝑁 − 𝑀) ∈ ℕ0) → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑀)) | |
34 | 1, 32, 8, 33 | syl3anc 1368 | . 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑀)) |
35 | 20, 34 | eqsstrrd 3954 | 1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 {cpr 4527 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑pm cpm 8390 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 − cmin 10859 ℕ0cn0 11885 ℤ≥cuz 12231 ...cfz 12885 D𝑛 cdvn 24467 |
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-inf2 9088 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 |
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-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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 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-seq 13365 df-exp 13426 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-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 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-cnp 21833 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-limc 24469 df-dv 24470 df-dvn 24471 |
This theorem is referenced by: taylplem1 24958 taylply2 24963 taylply 24964 taylthlem1 24968 taylthlem2 24969 |
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