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Mirrors > Home > MPE Home > Th. List > mat1ov | Structured version Visualization version GIF version |
Description: Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.) |
Ref | Expression |
---|---|
mat1.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mat1.o | ⊢ 1 = (1r‘𝑅) |
mat1.z | ⊢ 0 = (0g‘𝑅) |
mat1ov.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mat1ov.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mat1ov.i | ⊢ (𝜑 → 𝐼 ∈ 𝑁) |
mat1ov.j | ⊢ (𝜑 → 𝐽 ∈ 𝑁) |
mat1ov.u | ⊢ 𝑈 = (1r‘𝐴) |
Ref | Expression |
---|---|
mat1ov | ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mat1ov.u | . . 3 ⊢ 𝑈 = (1r‘𝐴) | |
2 | mat1ov.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
3 | mat1ov.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
4 | mat1.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | mat1.o | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
6 | mat1.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
7 | 4, 5, 6 | mat1 22362 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) |
8 | 2, 3, 7 | syl2anc 583 | . . 3 ⊢ (𝜑 → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) |
9 | 1, 8 | eqtrid 2780 | . 2 ⊢ (𝜑 → 𝑈 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) |
10 | eqeq12 2745 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖 = 𝑗 ↔ 𝐼 = 𝐽)) | |
11 | 10 | ifbid 4552 | . . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐼 = 𝐽, 1 , 0 )) |
12 | 11 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐼 = 𝐽, 1 , 0 )) |
13 | mat1ov.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
14 | mat1ov.j | . 2 ⊢ (𝜑 → 𝐽 ∈ 𝑁) | |
15 | 5 | fvexi 6911 | . . . 4 ⊢ 1 ∈ V |
16 | 6 | fvexi 6911 | . . . 4 ⊢ 0 ∈ V |
17 | 15, 16 | ifex 4579 | . . 3 ⊢ if(𝐼 = 𝐽, 1 , 0 ) ∈ V |
18 | 17 | a1i 11 | . 2 ⊢ (𝜑 → if(𝐼 = 𝐽, 1 , 0 ) ∈ V) |
19 | 9, 12, 13, 14, 18 | ovmpod 7573 | 1 ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ifcif 4529 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 Fincfn 8964 0gc0g 17421 1rcur 20121 Ringcrg 20173 Mat cmat 22320 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-subrg 20508 df-lmod 20745 df-lss 20816 df-sra 21058 df-rgmod 21059 df-dsmm 21666 df-frlm 21681 df-mamu 22299 df-mat 22321 |
This theorem is referenced by: dmatid 22410 scmatscmide 22422 ma1repveval 22486 1marepvmarrepid 22490 1marepvsma1 22498 mdet1 22516 mdetunilem8 22534 pmat1ovd 22612 mat2pmat1 22647 chpmat1dlem 22750 chpdmatlem2 22754 chpdmatlem3 22755 chpidmat 22762 1smat1 33405 matunitlindflem2 37090 |
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