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Mirrors > Home > MPE Home > Th. List > 1pmatscmul | Structured version Visualization version GIF version |
Description: The scalar product of the identity polynomial matrix with a polynomial is a polynomial matrix. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.) |
Ref | Expression |
---|---|
1pmatscmul.p | ⊢ 𝑃 = (Poly1‘𝑅) |
1pmatscmul.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
1pmatscmul.b | ⊢ 𝐵 = (Base‘𝐶) |
1pmatscmul.e | ⊢ 𝐸 = (Base‘𝑃) |
1pmatscmul.m | ⊢ ∗ = ( ·𝑠 ‘𝐶) |
1pmatscmul.1 | ⊢ 1 = (1r‘𝐶) |
Ref | Expression |
---|---|
1pmatscmul | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑄 ∗ 1 ) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pmatscmul.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1ring 21615 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
3 | 2 | anim2i 617 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
4 | 3 | 3adant3 1132 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
5 | simp3 1138 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸) | |
6 | 1pmatscmul.c | . . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
7 | 1, 6 | pmatring 22037 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
8 | 7 | 3adant3 1132 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → 𝐶 ∈ Ring) |
9 | 1pmatscmul.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
10 | 1pmatscmul.1 | . . . 4 ⊢ 1 = (1r‘𝐶) | |
11 | 9, 10 | ringidcl 19985 | . . 3 ⊢ (𝐶 ∈ Ring → 1 ∈ 𝐵) |
12 | 8, 11 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → 1 ∈ 𝐵) |
13 | 1pmatscmul.e | . . 3 ⊢ 𝐸 = (Base‘𝑃) | |
14 | 1pmatscmul.m | . . 3 ⊢ ∗ = ( ·𝑠 ‘𝐶) | |
15 | 13, 6, 9, 14 | matvscl 21776 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ (𝑄 ∈ 𝐸 ∧ 1 ∈ 𝐵)) → (𝑄 ∗ 1 ) ∈ 𝐵) |
16 | 4, 5, 12, 15 | syl12anc 835 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑄 ∗ 1 ) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6494 (class class class)co 7354 Fincfn 8880 Basecbs 17080 ·𝑠 cvsca 17134 1rcur 19909 Ringcrg 19960 Poly1cpl1 21544 Mat cmat 21750 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-ofr 7615 df-om 7800 df-1st 7918 df-2nd 7919 df-supp 8090 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-map 8764 df-pm 8765 df-ixp 8833 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-fsupp 9303 df-sup 9375 df-oi 9443 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-fz 13422 df-fzo 13565 df-seq 13904 df-hash 14228 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-sca 17146 df-vsca 17147 df-ip 17148 df-tset 17149 df-ple 17150 df-ds 17152 df-hom 17154 df-cco 17155 df-0g 17320 df-gsum 17321 df-prds 17326 df-pws 17328 df-mre 17463 df-mrc 17464 df-acs 17466 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-mhm 18598 df-submnd 18599 df-grp 18748 df-minusg 18749 df-sbg 18750 df-mulg 18869 df-subg 18921 df-ghm 19002 df-cntz 19093 df-cmn 19560 df-abl 19561 df-mgp 19893 df-ur 19910 df-ring 19962 df-subrg 20216 df-lmod 20320 df-lss 20389 df-sra 20629 df-rgmod 20630 df-dsmm 21134 df-frlm 21149 df-psr 21307 df-mpl 21309 df-opsr 21311 df-psr1 21547 df-ply1 21549 df-mamu 21729 df-mat 21751 |
This theorem is referenced by: pmatcollpwscmatlem2 22135 pmatcollpwscmat 22136 |
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