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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 21296 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌)) |
6 | 5 | oveqd 7219 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐼(𝑋 ✚ 𝑌)𝐽) = (𝐼(𝑋 ∘f + 𝑌)𝐽)) |
7 | 6 | adantr 484 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ✚ 𝑌)𝐽) = (𝐼(𝑋 ∘f + 𝑌)𝐽)) |
8 | df-ov 7205 | . . 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 5576 | . . 3 ⊢ (〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁) ↔ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) | |
11 | eqid 2734 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
12 | 1, 11, 2 | matbas2i 21291 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
13 | elmapfn 8535 | . . . . . 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 484 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 Fn (𝑁 × 𝑁)) |
16 | 1, 11, 2 | matbas2i 21291 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
17 | elmapfn 8535 | . . . . . 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 485 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 Fn (𝑁 × 𝑁)) |
20 | 1, 2 | matrcl 21281 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
21 | xpfi 8931 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) | |
22 | 21 | anidms 570 | . . . . . . 7 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
23 | 22 | adantr 484 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 × 𝑁) ∈ Fin) |
24 | 20, 23 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝑁 × 𝑁) ∈ Fin) |
25 | 24 | adantr 484 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁 × 𝑁) ∈ Fin) |
26 | inidm 4123 | . . . 4 ⊢ ((𝑁 × 𝑁) ∩ (𝑁 × 𝑁)) = (𝑁 × 𝑁) | |
27 | df-ov 7205 | . . . . . 6 ⊢ (𝐼𝑋𝐽) = (𝑋‘〈𝐼, 𝐽〉) | |
28 | 27 | eqcomi 2743 | . . . . 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 7205 | . . . . . 6 ⊢ (𝐼𝑌𝐽) = (𝑌‘〈𝐼, 𝐽〉) | |
31 | 30 | eqcomi 2743 | . . . . 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 7468 | . . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁)) → ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
34 | 10, 33 | sylan2br 598 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((𝑋 ∘f + 𝑌)‘〈𝐼, 𝐽〉) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) |
35 | 7, 9, 34 | 3eqtrd 2778 | 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 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 〈cop 4537 × cxp 5538 Fn wfn 6364 ‘cfv 6369 (class class class)co 7202 ∘f cof 7456 ↑m cmap 8497 Fincfn 8615 Basecbs 16684 +gcplusg 16767 Mat cmat 21276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-ot 4540 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-prds 16924 df-pws 16926 df-sra 20181 df-rgmod 20182 df-dsmm 20666 df-frlm 20681 df-mat 21277 |
This theorem is referenced by: mat1ghm 21352 cpmatacl 21585 mat2pmatghm 21599 pm2mpghm 21685 |
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