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Mirrors > Home > MPE Home > Th. List > matplusgcell | Structured version Visualization version GIF version |
Description: Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.) |
Ref | Expression |
---|---|
matplusgcell.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matplusgcell.b | ⊢ 𝐵 = (Base‘𝐴) |
matplusgcell.p | ⊢ ✚ = (+g‘𝐴) |
matplusgcell.q | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
matplusgcell | ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ✚ 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matplusgcell.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | matplusgcell.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
3 | matplusgcell.p | . . . . 5 ⊢ ✚ = (+g‘𝐴) | |
4 | matplusgcell.q | . . . . 5 ⊢ + = (+g‘𝑅) | |
5 | 1, 2, 3, 4 | matplusg2 21674 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌)) |
6 | 5 | oveqd 7346 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐼(𝑋 ✚ 𝑌)𝐽) = (𝐼(𝑋 ∘f + 𝑌)𝐽)) |
7 | 6 | adantr 481 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ✚ 𝑌)𝐽) = (𝐼(𝑋 ∘f + 𝑌)𝐽)) |
8 | df-ov 7332 | . . 3 ⊢ (𝐼(𝑋 ∘f + 𝑌)𝐽) = ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉) | |
9 | 8 | a1i 11 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ∘f + 𝑌)𝐽) = ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉)) |
10 | opelxp 5650 | . . 3 ⊢ (〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁) ↔ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) | |
11 | eqid 2736 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
12 | 1, 11, 2 | matbas2i 21669 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
13 | elmapfn 8716 | . . . . . 6 ⊢ (𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑋 Fn (𝑁 × 𝑁)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 Fn (𝑁 × 𝑁)) |
15 | 14 | adantr 481 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 Fn (𝑁 × 𝑁)) |
16 | 1, 11, 2 | matbas2i 21669 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
17 | elmapfn 8716 | . . . . . 6 ⊢ (𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑌 Fn (𝑁 × 𝑁)) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝑌 ∈ 𝐵 → 𝑌 Fn (𝑁 × 𝑁)) |
19 | 18 | adantl 482 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 Fn (𝑁 × 𝑁)) |
20 | 1, 2 | matrcl 21657 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
21 | xpfi 9174 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) | |
22 | 21 | anidms 567 | . . . . . . 7 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
23 | 22 | adantr 481 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 × 𝑁) ∈ Fin) |
24 | 20, 23 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝑁 × 𝑁) ∈ Fin) |
25 | 24 | adantr 481 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁 × 𝑁) ∈ Fin) |
26 | inidm 4164 | . . . 4 ⊢ ((𝑁 × 𝑁) ∩ (𝑁 × 𝑁)) = (𝑁 × 𝑁) | |
27 | df-ov 7332 | . . . . . 6 ⊢ (𝐼𝑋𝐽) = (𝑋‘〈𝐼, 𝐽〉) | |
28 | 27 | eqcomi 2745 | . . . . 5 ⊢ (𝑋‘〈𝐼, 𝐽〉) = (𝐼𝑋𝐽) |
29 | 28 | a1i 11 | . . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) → (𝑋‘〈𝐼, 𝐽〉) = (𝐼𝑋𝐽)) |
30 | df-ov 7332 | . . . . . 6 ⊢ (𝐼𝑌𝐽) = (𝑌‘〈𝐼, 𝐽〉) | |
31 | 30 | eqcomi 2745 | . . . . 5 ⊢ (𝑌‘〈𝐼, 𝐽〉) = (𝐼𝑌𝐽) |
32 | 31 | a1i 11 | . . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) → (𝑌‘〈𝐼, 𝐽〉) = (𝐼𝑌𝐽)) |
33 | 15, 19, 25, 25, 26, 29, 32 | ofval 7598 | . . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) → ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
34 | 10, 33 | sylan2br 595 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
35 | 7, 9, 34 | 3eqtrd 2780 | 1 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ✚ 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 〈cop 4578 × cxp 5612 Fn wfn 6468 ‘cfv 6473 (class class class)co 7329 ∘f cof 7585 ↑m cmap 8678 Fincfn 8796 Basecbs 17001 +gcplusg 17051 Mat cmat 21652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-ot 4581 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-sup 9291 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-fz 13333 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-hom 17075 df-cco 17076 df-0g 17241 df-prds 17247 df-pws 17249 df-sra 20532 df-rgmod 20533 df-dsmm 21037 df-frlm 21052 df-mat 21653 |
This theorem is referenced by: mat1ghm 21730 cpmatacl 21963 mat2pmatghm 21977 pm2mpghm 22063 |
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