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Mirrors > Home > MPE Home > Th. List > ply1scln0 | Structured version Visualization version GIF version |
Description: Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
ply1scl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1scl.a | ⊢ 𝐴 = (algSc‘𝑃) |
ply1scl0.z | ⊢ 0 = (0g‘𝑅) |
ply1scl0.y | ⊢ 𝑌 = (0g‘𝑃) |
ply1scln0.k | ⊢ 𝐾 = (Base‘𝑅) |
Ref | Expression |
---|---|
ply1scln0 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → (𝐴‘𝑋) ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1scl.p | . . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | ply1scl.a | . . . . . . . 8 ⊢ 𝐴 = (algSc‘𝑃) | |
3 | ply1scln0.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅) | |
4 | eqid 2651 | . . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | 1, 2, 3, 4 | ply1sclf1 19707 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐴:𝐾–1-1→(Base‘𝑃)) |
6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → 𝐴:𝐾–1-1→(Base‘𝑃)) |
7 | simpr 476 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
8 | ply1scl0.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
9 | 3, 8 | ring0cl 18615 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
10 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → 0 ∈ 𝐾) |
11 | f1fveq 6559 | . . . . . 6 ⊢ ((𝐴:𝐾–1-1→(Base‘𝑃) ∧ (𝑋 ∈ 𝐾 ∧ 0 ∈ 𝐾)) → ((𝐴‘𝑋) = (𝐴‘ 0 ) ↔ 𝑋 = 0 )) | |
12 | 6, 7, 10, 11 | syl12anc 1364 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → ((𝐴‘𝑋) = (𝐴‘ 0 ) ↔ 𝑋 = 0 )) |
13 | 12 | biimpd 219 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → ((𝐴‘𝑋) = (𝐴‘ 0 ) → 𝑋 = 0 )) |
14 | 13 | necon3d 2844 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝑋 ≠ 0 → (𝐴‘𝑋) ≠ (𝐴‘ 0 ))) |
15 | 14 | 3impia 1280 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → (𝐴‘𝑋) ≠ (𝐴‘ 0 )) |
16 | ply1scl0.y | . . . 4 ⊢ 𝑌 = (0g‘𝑃) | |
17 | 1, 2, 8, 16 | ply1scl0 19708 | . . 3 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = 𝑌) |
18 | 17 | 3ad2ant1 1102 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → (𝐴‘ 0 ) = 𝑌) |
19 | 15, 18 | neeqtrd 2892 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → (𝐴‘𝑋) ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 –1-1→wf1 5923 ‘cfv 5926 Basecbs 15904 0gc0g 16147 Ringcrg 18593 algSccascl 19359 Poly1cpl1 19595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-ofr 6940 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-fzo 12505 df-seq 12842 df-hash 13158 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-tset 16007 df-ple 16008 df-0g 16149 df-gsum 16150 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-ghm 17705 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-subrg 18826 df-lmod 18913 df-lss 18981 df-ascl 19362 df-psr 19404 df-mvr 19405 df-mpl 19406 df-opsr 19408 df-psr1 19598 df-vr1 19599 df-ply1 19600 df-coe1 19601 |
This theorem is referenced by: deg1scl 23918 ply1nz 23926 |
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