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Mirrors > Home > MPE Home > Th. List > mat1dim0 | Structured version Visualization version GIF version |
Description: The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.) |
Ref | Expression |
---|---|
mat1dim.a | ⊢ 𝐴 = ({𝐸} Mat 𝑅) |
mat1dim.b | ⊢ 𝐵 = (Base‘𝑅) |
mat1dim.o | ⊢ 𝑂 = 〈𝐸, 𝐸〉 |
Ref | Expression |
---|---|
mat1dim0 | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (0g‘𝐴) = {〈𝑂, (0g‘𝑅)〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snfi 8988 | . . . . . 6 ⊢ {𝐸} ∈ Fin | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → {𝐸} ∈ Fin) |
3 | 2 | anim2i 617 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑅 ∈ Ring ∧ {𝐸} ∈ Fin)) |
4 | 3 | ancomd 462 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring)) |
5 | mat1dim.a | . . . 4 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
6 | eqid 2736 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 5, 6 | mat0op 21768 | . . 3 ⊢ (({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅))) |
8 | 4, 7 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (0g‘𝐴) = (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅))) |
9 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
10 | fvexd 6857 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (0g‘𝑅) ∈ V) | |
11 | eqid 2736 | . . . . 5 ⊢ (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅)) = (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅)) | |
12 | eqidd 2737 | . . . . 5 ⊢ (𝑥 = 𝐸 → (0g‘𝑅) = (0g‘𝑅)) | |
13 | eqidd 2737 | . . . . 5 ⊢ (𝑦 = 𝐸 → (0g‘𝑅) = (0g‘𝑅)) | |
14 | 11, 12, 13 | mposn 8035 | . . . 4 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ (0g‘𝑅) ∈ V) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅)) = {〈〈𝐸, 𝐸〉, (0g‘𝑅)〉}) |
15 | mat1dim.o | . . . . . . 7 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
16 | 15 | eqcomi 2745 | . . . . . 6 ⊢ 〈𝐸, 𝐸〉 = 𝑂 |
17 | 16 | opeq1i 4833 | . . . . 5 ⊢ 〈〈𝐸, 𝐸〉, (0g‘𝑅)〉 = 〈𝑂, (0g‘𝑅)〉 |
18 | 17 | sneqi 4597 | . . . 4 ⊢ {〈〈𝐸, 𝐸〉, (0g‘𝑅)〉} = {〈𝑂, (0g‘𝑅)〉} |
19 | 14, 18 | eqtrdi 2792 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ (0g‘𝑅) ∈ V) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅)) = {〈𝑂, (0g‘𝑅)〉}) |
20 | 9, 9, 10, 19 | syl3anc 1371 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅)) = {〈𝑂, (0g‘𝑅)〉}) |
21 | 8, 20 | eqtrd 2776 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (0g‘𝐴) = {〈𝑂, (0g‘𝑅)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3445 {csn 4586 〈cop 4592 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 Fincfn 8883 Basecbs 17083 0gc0g 17321 Ringcrg 19964 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-rmo 3353 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-om 7803 df-1st 7921 df-2nd 7922 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-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-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-mgp 19897 df-ur 19914 df-ring 19966 df-subrg 20220 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-dsmm 21138 df-frlm 21153 df-mat 21755 |
This theorem is referenced by: (None) |
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