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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 8918 | . . . . . 6 ⊢ {𝐸} ∈ Fin | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → {𝐸} ∈ Fin) |
3 | 2 | anim2i 618 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑅 ∈ Ring ∧ {𝐸} ∈ Fin)) |
4 | 3 | ancomd 463 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring)) |
5 | mat1dim.a | . . . 4 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
6 | eqid 2737 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 5, 6 | mat0op 21678 | . . 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 486 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
10 | fvexd 6849 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (0g‘𝑅) ∈ V) | |
11 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅)) = (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ (0g‘𝑅)) | |
12 | eqidd 2738 | . . . . 5 ⊢ (𝑥 = 𝐸 → (0g‘𝑅) = (0g‘𝑅)) | |
13 | eqidd 2738 | . . . . 5 ⊢ (𝑦 = 𝐸 → (0g‘𝑅) = (0g‘𝑅)) | |
14 | 11, 12, 13 | mposn 8020 | . . . 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 2746 | . . . . . 6 ⊢ 〈𝐸, 𝐸〉 = 𝑂 |
17 | 16 | opeq1i 4828 | . . . . 5 ⊢ 〈〈𝐸, 𝐸〉, (0g‘𝑅)〉 = 〈𝑂, (0g‘𝑅)〉 |
18 | 17 | sneqi 4592 | . . . 4 ⊢ {〈〈𝐸, 𝐸〉, (0g‘𝑅)〉} = {〈𝑂, (0g‘𝑅)〉} |
19 | 14, 18 | eqtrdi 2793 | . . 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 2777 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (0g‘𝐴) = {〈𝑂, (0g‘𝑅)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3443 {csn 4581 〈cop 4587 ‘cfv 6488 (class class class)co 7346 ∈ cmpo 7348 Fincfn 8813 Basecbs 17014 0gc0g 17252 Ringcrg 19882 Mat cmat 21664 |
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 2708 ax-rep 5237 ax-sep 5251 ax-nul 5258 ax-pow 5315 ax-pr 5379 ax-un 7659 ax-cnex 11037 ax-resscn 11038 ax-1cn 11039 ax-icn 11040 ax-addcl 11041 ax-addrcl 11042 ax-mulcl 11043 ax-mulrcl 11044 ax-mulcom 11045 ax-addass 11046 ax-mulass 11047 ax-distr 11048 ax-i2m1 11049 ax-1ne0 11050 ax-1rid 11051 ax-rnegex 11052 ax-rrecex 11053 ax-cnre 11054 ax-pre-lttri 11055 ax-pre-lttrn 11056 ax-pre-ltadd 11057 ax-pre-mulgt0 11058 |
This theorem depends on definitions: df-bi 206 df-an 398 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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3735 df-csb 3851 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3924 df-nul 4278 df-if 4482 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4861 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5184 df-tr 5218 df-id 5525 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5582 df-we 5584 df-xp 5633 df-rel 5634 df-cnv 5635 df-co 5636 df-dm 5637 df-rn 5638 df-res 5639 df-ima 5640 df-pred 6246 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6440 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7302 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7790 df-1st 7908 df-2nd 7909 df-frecs 8176 df-wrecs 8207 df-recs 8281 df-rdg 8320 df-1o 8376 df-er 8578 df-map 8697 df-ixp 8766 df-en 8814 df-dom 8815 df-sdom 8816 df-fin 8817 df-sup 9308 df-pnf 11121 df-mnf 11122 df-xr 11123 df-ltxr 11124 df-le 11125 df-sub 11317 df-neg 11318 df-nn 12084 df-2 12146 df-3 12147 df-4 12148 df-5 12149 df-6 12150 df-7 12151 df-8 12152 df-9 12153 df-n0 12344 df-z 12430 df-dec 12548 df-uz 12693 df-fz 13350 df-struct 16950 df-sets 16967 df-slot 16985 df-ndx 16997 df-base 17015 df-ress 17044 df-plusg 17077 df-mulr 17078 df-sca 17080 df-vsca 17081 df-ip 17082 df-tset 17083 df-ple 17084 df-ds 17086 df-hom 17088 df-cco 17089 df-0g 17254 df-prds 17260 df-pws 17262 df-mgm 18428 df-sgrp 18477 df-mnd 18488 df-grp 18681 df-minusg 18682 df-sbg 18683 df-subg 18853 df-mgp 19820 df-ur 19837 df-ring 19884 df-subrg 20131 df-lmod 20235 df-lss 20304 df-sra 20544 df-rgmod 20545 df-dsmm 21049 df-frlm 21064 df-mat 21665 |
This theorem is referenced by: (None) |
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