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Mirrors > Home > MPE Home > Th. List > mat1dimid | Structured version Visualization version GIF version |
Description: The identity 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 |
---|---|
mat1dimid | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (1r‘𝐴) = {〈𝑂, (1r‘𝑅)〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snfi 9081 | . . . . . 6 ⊢ {𝐸} ∈ Fin | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → {𝐸} ∈ Fin) |
3 | 2 | anim2i 615 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑅 ∈ Ring ∧ {𝐸} ∈ Fin)) |
4 | 3 | ancomd 460 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring)) |
5 | mat1dim.a | . . . 4 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
6 | eqid 2726 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | eqid 2726 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | 5, 6, 7 | mat1 22440 | . . 3 ⊢ (({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅)))) |
9 | 4, 8 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (1r‘𝐴) = (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅)))) |
10 | simpr 483 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
11 | fvex 6914 | . . . . . . 7 ⊢ (1r‘𝑅) ∈ V | |
12 | fvex 6914 | . . . . . . 7 ⊢ (0g‘𝑅) ∈ V | |
13 | 11, 12 | ifex 4583 | . . . . . 6 ⊢ if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅)) ∈ V |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅)) ∈ V) |
15 | eqid 2726 | . . . . . 6 ⊢ (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅))) = (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅))) | |
16 | eqeq1 2730 | . . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑥 = 𝑦 ↔ 𝐸 = 𝑦)) | |
17 | 16 | ifbid 4556 | . . . . . 6 ⊢ (𝑥 = 𝐸 → if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅)) = if(𝐸 = 𝑦, (1r‘𝑅), (0g‘𝑅))) |
18 | eqeq2 2738 | . . . . . . 7 ⊢ (𝑦 = 𝐸 → (𝐸 = 𝑦 ↔ 𝐸 = 𝐸)) | |
19 | 18 | ifbid 4556 | . . . . . 6 ⊢ (𝑦 = 𝐸 → if(𝐸 = 𝑦, (1r‘𝑅), (0g‘𝑅)) = if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))) |
20 | 15, 17, 19 | mposn 8117 | . . . . 5 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅)) ∈ V) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅))) = {〈〈𝐸, 𝐸〉, if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))〉}) |
21 | 10, 10, 14, 20 | syl3anc 1368 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅))) = {〈〈𝐸, 𝐸〉, if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))〉}) |
22 | eqid 2726 | . . . . . . 7 ⊢ 𝐸 = 𝐸 | |
23 | 22 | iftruei 4540 | . . . . . 6 ⊢ if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅) |
24 | 23 | opeq2i 4883 | . . . . 5 ⊢ 〈〈𝐸, 𝐸〉, if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))〉 = 〈〈𝐸, 𝐸〉, (1r‘𝑅)〉 |
25 | 24 | sneqi 4644 | . . . 4 ⊢ {〈〈𝐸, 𝐸〉, if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))〉} = {〈〈𝐸, 𝐸〉, (1r‘𝑅)〉} |
26 | 21, 25 | eqtrdi 2782 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅))) = {〈〈𝐸, 𝐸〉, (1r‘𝑅)〉}) |
27 | mat1dim.o | . . . . 5 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
28 | 27 | opeq1i 4882 | . . . 4 ⊢ 〈𝑂, (1r‘𝑅)〉 = 〈〈𝐸, 𝐸〉, (1r‘𝑅)〉 |
29 | 28 | sneqi 4644 | . . 3 ⊢ {〈𝑂, (1r‘𝑅)〉} = {〈〈𝐸, 𝐸〉, (1r‘𝑅)〉} |
30 | 26, 29 | eqtr4di 2784 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅))) = {〈𝑂, (1r‘𝑅)〉}) |
31 | 9, 30 | eqtrd 2766 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (1r‘𝐴) = {〈𝑂, (1r‘𝑅)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ifcif 4533 {csn 4633 〈cop 4639 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 Fincfn 8974 Basecbs 17213 0gc0g 17454 1rcur 20164 Ringcrg 20216 Mat cmat 22398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-ot 4642 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-subrg 20553 df-lmod 20838 df-lss 20909 df-sra 21151 df-rgmod 21152 df-dsmm 21730 df-frlm 21745 df-mamu 22382 df-mat 22399 |
This theorem is referenced by: mat1mhm 22477 mat1scmat 22532 |
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