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Mirrors > Home > MPE Home > Th. List > matmulcell | Structured version Visualization version GIF version |
Description: Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) |
Ref | Expression |
---|---|
matmulcell.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matmulcell.b | ⊢ 𝐵 = (Base‘𝐴) |
matmulcell.t | ⊢ · = (.r‘𝑅) |
matmulcell.m | ⊢ × = (.r‘𝐴) |
Ref | Expression |
---|---|
matmulcell | ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝐽))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matmulcell.a | . . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | matmulcell.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐴) | |
3 | 1, 2 | matrcl 20266 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | eqid 2651 | . . . . . . . . . . 11 ⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
5 | 1, 4 | matmulr 20292 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
6 | matmulcell.m | . . . . . . . . . 10 ⊢ × = (.r‘𝐴) | |
7 | 5, 6 | syl6reqr 2704 | . . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)) |
8 | 7 | a1d 25 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑅 ∈ Ring → × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉))) |
9 | 3, 8 | syl 17 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (𝑅 ∈ Ring → × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉))) |
10 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ Ring → × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉))) |
11 | 10 | impcom 445 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)) |
12 | 11 | 3adant3 1101 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)) |
13 | 12 | oveqd 6707 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝑋 × 𝑌) = (𝑋(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑌)) |
14 | 13 | oveqd 6707 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝐼(𝑋(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑌)𝐽)) |
15 | eqid 2651 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | eqid 2651 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
17 | simp1 1081 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
18 | 3 | simpld 474 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑁 ∈ Fin) |
20 | 19 | 3ad2ant2 1103 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑁 ∈ Fin) |
21 | 1, 15, 2 | matbas2i 20276 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
23 | 22 | 3ad2ant2 1103 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑋 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
24 | 1, 15, 2 | matbas2i 20276 | . . . . 5 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
25 | 24 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
26 | 25 | 3ad2ant2 1103 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
27 | simp3l 1109 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁) | |
28 | simp3r 1110 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐽 ∈ 𝑁) | |
29 | 4, 15, 16, 17, 20, 20, 20, 23, 26, 27, 28 | mamufv 20241 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑌)𝐽) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝐽))))) |
30 | 14, 29 | eqtrd 2685 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝐽))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 Vcvv 3231 〈cotp 4218 ↦ cmpt 4762 × cxp 5141 ‘cfv 5926 (class class class)co 6690 ↑𝑚 cmap 7899 Fincfn 7997 Basecbs 15904 .rcmulr 15989 Σg cgsu 16148 Ringcrg 18593 maMul cmmul 20237 Mat cmat 20261 |
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-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-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-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 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-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-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 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-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-sup 8389 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-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-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-hom 16013 df-cco 16014 df-0g 16149 df-prds 16155 df-pws 16157 df-sra 19220 df-rgmod 19221 df-dsmm 20124 df-frlm 20139 df-mamu 20238 df-mat 20262 |
This theorem is referenced by: mat1mhm 20338 scmatscm 20367 cpmatmcl 20572 |
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