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Mirrors > Home > MPE Home > Th. List > mat0op | Structured version Visualization version GIF version |
Description: Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.) |
Ref | Expression |
---|---|
mat0op.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mat0op.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
mat0op | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mat0op.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | eqid 2739 | . . 3 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁)) | |
3 | 1, 2 | mat0 21547 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴)) |
4 | fconstmpo 7382 | . . 3 ⊢ ((𝑁 × 𝑁) × {(0g‘𝑅)}) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)) | |
5 | simpr 484 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring) | |
6 | sqxpexg 7596 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ V) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 × 𝑁) ∈ V) |
8 | eqid 2739 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | 2, 8 | frlm0 20942 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 × 𝑁) ∈ V) → ((𝑁 × 𝑁) × {(0g‘𝑅)}) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
10 | 5, 7, 9 | syl2anc 583 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 × 𝑁) × {(0g‘𝑅)}) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
11 | mat0op.z | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
12 | 11 | eqcomi 2748 | . . . . . 6 ⊢ (0g‘𝑅) = 0 |
13 | 12 | a1i 11 | . . . . 5 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (0g‘𝑅) = 0 ) |
14 | 13 | mpoeq3ia 7344 | . . . 4 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 ) |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 )) |
16 | 4, 10, 15 | 3eqtr3a 2803 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 )) |
17 | 3, 16 | eqtr3d 2781 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 {csn 4566 × cxp 5586 ‘cfv 6430 (class class class)co 7268 ∈ cmpo 7270 Fincfn 8707 0gc0g 17131 Ringcrg 19764 freeLMod cfrlm 20934 Mat cmat 21535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-hom 16967 df-cco 16968 df-0g 17133 df-prds 17139 df-pws 17141 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-mgp 19702 df-ur 19719 df-ring 19766 df-subrg 20003 df-lmod 20106 df-lss 20175 df-sra 20415 df-rgmod 20416 df-dsmm 20920 df-frlm 20935 df-mat 21536 |
This theorem is referenced by: matinvgcell 21565 mat1dim0 21603 mdet0 21736 pmat0op 21825 decpmataa0 21898 decpmatid 21900 decpmatmulsumfsupp 21903 pmatcollpw2lem 21907 monmatcollpw 21909 mptcoe1matfsupp 21932 mp2pm2mplem4 21939 pm2mpmhmlem1 21948 chp0mat 21976 |
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