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Mirrors > Home > MPE Home > Th. List > deg1ldgn | Structured version Visualization version GIF version |
Description: An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
deg1z.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1z.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1z.z | ⊢ 0 = (0g‘𝑃) |
deg1nn0cl.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1ldg.y | ⊢ 𝑌 = (0g‘𝑅) |
deg1ldg.a | ⊢ 𝐴 = (coe1‘𝐹) |
deg1ldgn.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
deg1ldgn.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
deg1ldgn.x | ⊢ (𝜑 → 𝑋 ∈ ℕ0) |
deg1ldgn.e | ⊢ (𝜑 → (𝐴‘𝑋) = 𝑌) |
Ref | Expression |
---|---|
deg1ldgn | ⊢ (𝜑 → (𝐷‘𝐹) ≠ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1ldgn.e | . 2 ⊢ (𝜑 → (𝐴‘𝑋) = 𝑌) | |
2 | fveq2 6897 | . . . . . 6 ⊢ ((𝐷‘𝐹) = 𝑋 → (𝐴‘(𝐷‘𝐹)) = (𝐴‘𝑋)) | |
3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → (𝐴‘(𝐷‘𝐹)) = (𝐴‘𝑋)) |
4 | deg1ldgn.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
5 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → 𝑅 ∈ Ring) |
6 | deg1ldgn.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → 𝐹 ∈ 𝐵) |
8 | deg1ldgn.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℕ0) | |
9 | eleq1a 2824 | . . . . . . . . 9 ⊢ (𝑋 ∈ ℕ0 → ((𝐷‘𝐹) = 𝑋 → (𝐷‘𝐹) ∈ ℕ0)) | |
10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((𝐷‘𝐹) = 𝑋 → (𝐷‘𝐹) ∈ ℕ0)) |
11 | 10 | imp 406 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → (𝐷‘𝐹) ∈ ℕ0) |
12 | deg1z.d | . . . . . . . . . 10 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
13 | deg1z.p | . . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘𝑅) | |
14 | deg1z.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑃) | |
15 | deg1nn0cl.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑃) | |
16 | 12, 13, 14, 15 | deg1nn0clb 26039 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ0)) |
17 | 4, 6, 16 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ0)) |
18 | 17 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ0)) |
19 | 11, 18 | mpbird 257 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → 𝐹 ≠ 0 ) |
20 | deg1ldg.y | . . . . . . 7 ⊢ 𝑌 = (0g‘𝑅) | |
21 | deg1ldg.a | . . . . . . 7 ⊢ 𝐴 = (coe1‘𝐹) | |
22 | 12, 13, 14, 15, 20, 21 | deg1ldg 26041 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ 𝑌) |
23 | 5, 7, 19, 22 | syl3anc 1369 | . . . . 5 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → (𝐴‘(𝐷‘𝐹)) ≠ 𝑌) |
24 | 3, 23 | eqnetrrd 3006 | . . . 4 ⊢ ((𝜑 ∧ (𝐷‘𝐹) = 𝑋) → (𝐴‘𝑋) ≠ 𝑌) |
25 | 24 | ex 412 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) = 𝑋 → (𝐴‘𝑋) ≠ 𝑌)) |
26 | 25 | necon2d 2960 | . 2 ⊢ (𝜑 → ((𝐴‘𝑋) = 𝑌 → (𝐷‘𝐹) ≠ 𝑋)) |
27 | 1, 26 | mpd 15 | 1 ⊢ (𝜑 → (𝐷‘𝐹) ≠ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ‘cfv 6548 ℕ0cn0 12503 Basecbs 17180 0gc0g 17421 Ringcrg 20173 Poly1cpl1 22096 coe1cco1 22097 deg1 cdg1 26000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-ur 20122 df-ring 20175 df-cring 20176 df-cnfld 21280 df-psr 21842 df-mpl 21844 df-opsr 21846 df-psr1 22099 df-ply1 22101 df-coe1 22102 df-mdeg 26001 df-deg1 26002 |
This theorem is referenced by: deg1sublt 26059 |
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