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Mirrors > Home > MPE Home > Th. List > matvscacell | Structured version Visualization version GIF version |
Description: Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.) |
Ref | Expression |
---|---|
matplusgcell.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matplusgcell.b | ⊢ 𝐵 = (Base‘𝐴) |
matvscacell.k | ⊢ 𝐾 = (Base‘𝑅) |
matvscacell.v | ⊢ · = ( ·𝑠 ‘𝐴) |
matvscacell.t | ⊢ × = (.r‘𝑅) |
Ref | Expression |
---|---|
matvscacell | ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matplusgcell.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | matplusgcell.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
3 | matvscacell.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
4 | matvscacell.v | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐴) | |
5 | matvscacell.t | . . . . 5 ⊢ × = (.r‘𝑅) | |
6 | eqid 2771 | . . . . 5 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁) | |
7 | 1, 2, 3, 4, 5, 6 | matvsca2 20756 | . . . 4 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)) |
8 | 7 | oveqd 6991 | . . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝐼(𝑋 · 𝑌)𝐽) = (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)𝐽)) |
9 | 8 | 3ad2ant2 1115 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)𝐽)) |
10 | df-ov 6977 | . . 3 ⊢ (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)𝐽) = ((((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)‘〈𝐼, 𝐽〉) | |
11 | 10 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)𝐽) = ((((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)‘〈𝐼, 𝐽〉)) |
12 | opelxpi 5440 | . . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) | |
13 | 12 | 3ad2ant3 1116 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) |
14 | 1, 2 | matrcl 20740 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
15 | 14 | simpld 487 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → 𝑁 ∈ Fin) |
16 | 15 | adantl 474 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑁 ∈ Fin) |
17 | 16 | 3ad2ant2 1115 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑁 ∈ Fin) |
18 | xpfi 8582 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) | |
19 | 17, 17, 18 | syl2anc 576 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝑁 × 𝑁) ∈ Fin) |
20 | simp2l 1180 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑋 ∈ 𝐾) | |
21 | 2 | eleq2i 2850 | . . . . . . . . 9 ⊢ (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (Base‘𝐴)) |
22 | 21 | biimpi 208 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘𝐴)) |
23 | 22 | adantl 474 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ (Base‘𝐴)) |
24 | 23 | 3ad2ant2 1115 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 ∈ (Base‘𝐴)) |
25 | simp1 1117 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
26 | eqid 2771 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
27 | 1, 26 | matbas2 20749 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
28 | 17, 25, 27 | syl2anc 576 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
29 | 24, 28 | eleqtrrd 2862 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
30 | elmapfn 8227 | . . . . 5 ⊢ (𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) → 𝑌 Fn (𝑁 × 𝑁)) | |
31 | 29, 30 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 Fn (𝑁 × 𝑁)) |
32 | df-ov 6977 | . . . . . 6 ⊢ (𝐼𝑌𝐽) = (𝑌‘〈𝐼, 𝐽〉) | |
33 | 32 | eqcomi 2780 | . . . . 5 ⊢ (𝑌‘〈𝐼, 𝐽〉) = (𝐼𝑌𝐽) |
34 | 33 | a1i 11 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) → (𝑌‘〈𝐼, 𝐽〉) = (𝐼𝑌𝐽)) |
35 | 19, 20, 31, 34 | ofc1 7248 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) → ((((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)‘〈𝐼, 𝐽〉) = (𝑋 × (𝐼𝑌𝐽))) |
36 | 13, 35 | mpdan 675 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((((𝑁 × 𝑁) × {𝑋}) ∘𝑓 × 𝑌)‘〈𝐼, 𝐽〉) = (𝑋 × (𝐼𝑌𝐽))) |
37 | 9, 11, 36 | 3eqtrd 2811 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 Vcvv 3408 {csn 4435 〈cop 4441 × cxp 5401 Fn wfn 6180 ‘cfv 6185 (class class class)co 6974 ∘𝑓 cof 7223 ↑𝑚 cmap 8204 Fincfn 8304 Basecbs 16337 .rcmulr 16420 ·𝑠 cvsca 16423 Ringcrg 19032 Mat cmat 20735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-sup 8699 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-hom 16443 df-cco 16444 df-0g 16569 df-prds 16575 df-pws 16577 df-sra 19678 df-rgmod 19679 df-dsmm 20593 df-frlm 20608 df-mat 20736 |
This theorem is referenced by: dmatscmcl 20831 scmatscmide 20835 scmatscm 20841 mat2pmatlin 21062 monmatcollpw 21106 pmatcollpwlem 21107 chpmat1dlem 21162 chpdmatlem2 21166 chpdmatlem3 21167 |
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