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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 21776 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌)) |
6 | 5 | oveqd 7374 | . . 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 7360 | . . 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 5669 | . . 3 ⊢ (〈𝐼, 𝐽〉 ∈ (𝑁 × 𝑁) ↔ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) | |
11 | eqid 2736 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
12 | 1, 11, 2 | matbas2i 21771 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
13 | elmapfn 8803 | . . . . . 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 21771 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
17 | elmapfn 8803 | . . . . . 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 21759 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
21 | xpfi 9261 | . . . . . . . 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 4178 | . . . 4 ⊢ ((𝑁 × 𝑁) ∩ (𝑁 × 𝑁)) = (𝑁 × 𝑁) | |
27 | df-ov 7360 | . . . . . 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 7360 | . . . . . 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 7628 | . . 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 1541 ∈ wcel 2106 Vcvv 3445 〈cop 4592 × cxp 5631 Fn wfn 6491 ‘cfv 6496 (class class class)co 7357 ∘f cof 7615 ↑m cmap 8765 Fincfn 8883 Basecbs 17083 +gcplusg 17133 Mat cmat 21754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-prds 17329 df-pws 17331 df-sra 20633 df-rgmod 20634 df-dsmm 21138 df-frlm 21153 df-mat 21755 |
This theorem is referenced by: mat1ghm 21832 cpmatacl 22065 mat2pmatghm 22079 pm2mpghm 22165 |
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