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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 9082 | . . . . . 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 461 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring)) |
5 | mat1dim.a | . . . 4 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
6 | eqid 2735 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | eqid 2735 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | 5, 6, 7 | mat1 22469 | . . 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 484 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
11 | fvex 6920 | . . . . . . 7 ⊢ (1r‘𝑅) ∈ V | |
12 | fvex 6920 | . . . . . . 7 ⊢ (0g‘𝑅) ∈ V | |
13 | 11, 12 | ifex 4581 | . . . . . 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 2735 | . . . . . 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 2739 | . . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑥 = 𝑦 ↔ 𝐸 = 𝑦)) | |
17 | 16 | ifbid 4554 | . . . . . 6 ⊢ (𝑥 = 𝐸 → if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅)) = if(𝐸 = 𝑦, (1r‘𝑅), (0g‘𝑅))) |
18 | eqeq2 2747 | . . . . . . 7 ⊢ (𝑦 = 𝐸 → (𝐸 = 𝑦 ↔ 𝐸 = 𝐸)) | |
19 | 18 | ifbid 4554 | . . . . . 6 ⊢ (𝑦 = 𝐸 → if(𝐸 = 𝑦, (1r‘𝑅), (0g‘𝑅)) = if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))) |
20 | 15, 17, 19 | mposn 8127 | . . . . 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 1370 | . . . 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 2735 | . . . . . . 7 ⊢ 𝐸 = 𝐸 | |
23 | 22 | iftruei 4538 | . . . . . 6 ⊢ if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅) |
24 | 23 | opeq2i 4882 | . . . . 5 ⊢ 〈〈𝐸, 𝐸〉, if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))〉 = 〈〈𝐸, 𝐸〉, (1r‘𝑅)〉 |
25 | 24 | sneqi 4642 | . . . 4 ⊢ {〈〈𝐸, 𝐸〉, if(𝐸 = 𝐸, (1r‘𝑅), (0g‘𝑅))〉} = {〈〈𝐸, 𝐸〉, (1r‘𝑅)〉} |
26 | 21, 25 | eqtrdi 2791 | . . 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 4881 | . . . 4 ⊢ 〈𝑂, (1r‘𝑅)〉 = 〈〈𝐸, 𝐸〉, (1r‘𝑅)〉 |
29 | 28 | sneqi 4642 | . . 3 ⊢ {〈𝑂, (1r‘𝑅)〉} = {〈〈𝐸, 𝐸〉, (1r‘𝑅)〉} |
30 | 26, 29 | eqtr4di 2793 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑥 ∈ {𝐸}, 𝑦 ∈ {𝐸} ↦ if(𝑥 = 𝑦, (1r‘𝑅), (0g‘𝑅))) = {〈𝑂, (1r‘𝑅)〉}) |
31 | 9, 30 | eqtrd 2775 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (1r‘𝐴) = {〈𝑂, (1r‘𝑅)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ifcif 4531 {csn 4631 〈cop 4637 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8984 Basecbs 17245 0gc0g 17486 1rcur 20199 Ringcrg 20251 Mat cmat 22427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 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 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrg 20587 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-mamu 22411 df-mat 22428 |
This theorem is referenced by: mat1mhm 22506 mat1scmat 22561 |
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