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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 20964 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌)) |
6 | 5 | oveqd 7162 | . . 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 7148 | . . 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 5584 | . . 3 ⊢ (〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁) ↔ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) | |
11 | eqid 2818 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
12 | 1, 11, 2 | matbas2i 20959 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
13 | elmapfn 8418 | . . . . . 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 20959 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
17 | elmapfn 8418 | . . . . . 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 20949 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
21 | xpfi 8777 | . . . . . . . 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 4192 | . . . 4 ⊢ ((𝑁 × 𝑁) ∩ (𝑁 × 𝑁)) = (𝑁 × 𝑁) | |
27 | df-ov 7148 | . . . . . 6 ⊢ (𝐼𝑋𝐽) = (𝑋‘〈𝐼, 𝐽〉) | |
28 | 27 | eqcomi 2827 | . . . . 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 7148 | . . . . . 6 ⊢ (𝐼𝑌𝐽) = (𝑌‘〈𝐼, 𝐽〉) | |
31 | 30 | eqcomi 2827 | . . . . 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 7407 | . . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) → ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
34 | 10, 33 | sylan2br 594 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
35 | 7, 9, 34 | 3eqtrd 2857 | 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 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 × cxp 5546 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 ↑m cmap 8395 Fincfn 8497 Basecbs 16471 +gcplusg 16553 Mat cmat 20944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-sra 19873 df-rgmod 19874 df-dsmm 20804 df-frlm 20819 df-mat 20945 |
This theorem is referenced by: mat1ghm 21020 cpmatacl 21252 mat2pmatghm 21266 pm2mpghm 21352 |
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