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Mirrors > Home > MPE Home > Th. List > dgrsub | Structured version Visualization version GIF version |
Description: The degree of a difference of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
dgrsub.1 | ⊢ 𝑀 = (deg‘𝐹) |
dgrsub.2 | ⊢ 𝑁 = (deg‘𝐺) |
Ref | Expression |
---|---|
dgrsub | ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘𝑓 − 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyssc 23760 | . . . 4 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | 1 | sseli 3564 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
3 | ssid 3587 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
4 | neg1cn 11001 | . . . . 5 ⊢ -1 ∈ ℂ | |
5 | plyconst 23766 | . . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ -1 ∈ ℂ) → (ℂ × {-1}) ∈ (Poly‘ℂ)) | |
6 | 3, 4, 5 | mp2an 704 | . . . 4 ⊢ (ℂ × {-1}) ∈ (Poly‘ℂ) |
7 | 1 | sseli 3564 | . . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ)) |
8 | plymulcl 23781 | . . . 4 ⊢ (((ℂ × {-1}) ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ)) | |
9 | 6, 7, 8 | sylancr 694 | . . 3 ⊢ (𝐺 ∈ (Poly‘𝑆) → ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ)) |
10 | dgrsub.1 | . . . 4 ⊢ 𝑀 = (deg‘𝐹) | |
11 | eqid 2610 | . . . 4 ⊢ (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)) | |
12 | 10, 11 | dgradd 23827 | . . 3 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ)) → (deg‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) ≤ if(𝑀 ≤ (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)), (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)), 𝑀)) |
13 | 2, 9, 12 | syl2an 493 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) ≤ if(𝑀 ≤ (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)), (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)), 𝑀)) |
14 | plyf 23758 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
15 | plyf 23758 | . . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | |
16 | cnex 9896 | . . . . 5 ⊢ ℂ ∈ V | |
17 | ofnegsub 10895 | . . . . 5 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) | |
18 | 16, 17 | mp3an1 1403 | . . . 4 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) |
19 | 14, 15, 18 | syl2an 493 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) |
20 | 19 | fveq2d 6107 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (deg‘(𝐹 ∘𝑓 − 𝐺))) |
21 | neg1ne0 11003 | . . . . . . 7 ⊢ -1 ≠ 0 | |
22 | dgrmulc 23831 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = (deg‘𝐺)) | |
23 | 4, 21, 22 | mp3an12 1406 | . . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = (deg‘𝐺)) |
24 | dgrsub.2 | . . . . . 6 ⊢ 𝑁 = (deg‘𝐺) | |
25 | 23, 24 | syl6eqr 2662 | . . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = 𝑁) |
26 | 25 | adantl 481 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = 𝑁) |
27 | 26 | breq2d 4595 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 ≤ (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)) ↔ 𝑀 ≤ 𝑁)) |
28 | 27, 26 | ifbieq1d 4059 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀 ≤ (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)), (deg‘((ℂ × {-1}) ∘𝑓 · 𝐺)), 𝑀) = if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) |
29 | 13, 20, 28 | 3brtr3d 4614 | 1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘𝑓 − 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ifcif 4036 {csn 4125 class class class wbr 4583 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 -cneg 10146 Polycply 23744 degcdgr 23747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-0p 23243 df-ply 23748 df-coe 23750 df-dgr 23751 |
This theorem is referenced by: dgrcolem2 23834 plydivlem4 23855 plydiveu 23857 dgrsub2 36724 |
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