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Mirrors > Home > MPE Home > Th. List > ply10s0 | Structured version Visualization version GIF version |
Description: Zero times a univariate polynomial is the zero polynomial (lmod0vs 20184 analog.) (Contributed by AV, 2-Dec-2019.) |
Ref | Expression |
---|---|
ply10s0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply10s0.b | ⊢ 𝐵 = (Base‘𝑃) |
ply10s0.m | ⊢ ∗ = ( ·𝑠 ‘𝑃) |
ply10s0.e | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
ply10s0 | ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ( 0 ∗ 𝑀) = (0g‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply10s0.e | . . . 4 ⊢ 0 = (0g‘𝑅) | |
2 | ply10s0.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1sca 21452 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃)) |
5 | 4 | fveq2d 6796 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝑅) = (0g‘(Scalar‘𝑃))) |
6 | 1, 5 | eqtrid 2785 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 = (0g‘(Scalar‘𝑃))) |
7 | 6 | oveq1d 7310 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ( 0 ∗ 𝑀) = ((0g‘(Scalar‘𝑃)) ∗ 𝑀)) |
8 | 2 | ply1lmod 21451 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
9 | ply10s0.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
10 | eqid 2733 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
11 | ply10s0.m | . . . 4 ⊢ ∗ = ( ·𝑠 ‘𝑃) | |
12 | eqid 2733 | . . . 4 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃)) | |
13 | eqid 2733 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
14 | 9, 10, 11, 12, 13 | lmod0vs 20184 | . . 3 ⊢ ((𝑃 ∈ LMod ∧ 𝑀 ∈ 𝐵) → ((0g‘(Scalar‘𝑃)) ∗ 𝑀) = (0g‘𝑃)) |
15 | 8, 14 | sylan 579 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((0g‘(Scalar‘𝑃)) ∗ 𝑀) = (0g‘𝑃)) |
16 | 7, 15 | eqtrd 2773 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ( 0 ∗ 𝑀) = (0g‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 Scalarcsca 16993 ·𝑠 cvsca 16994 0gc0g 17178 Ringcrg 19811 LModclmod 20151 Poly1cpl1 21376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-sca 17006 df-vsca 17007 df-tset 17009 df-ple 17010 df-0g 17180 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-grp 18608 df-minusg 18609 df-sbg 18610 df-subg 18780 df-mgp 19749 df-ur 19766 df-ring 19813 df-lmod 20153 df-lss 20222 df-psr 21140 df-mpl 21142 df-opsr 21144 df-psr1 21379 df-ply1 21381 |
This theorem is referenced by: pmatcollpw1lem1 21951 pmatcollpw2lem 21954 |
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