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Mirrors > Home > MPE Home > Th. List > pmat1ovscd | Structured version Visualization version GIF version |
Description: Entries of the identity polynomial matrix over a ring represented with "lifted scalars", deduction form. (Contributed by AV, 16-Nov-2019.) |
Ref | Expression |
---|---|
pmat0opsc.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pmat0opsc.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pmat0opsc.a | ⊢ 𝐴 = (algSc‘𝑃) |
pmat0opsc.z | ⊢ 0 = (0g‘𝑅) |
pmat1opsc.o | ⊢ 1 = (1r‘𝑅) |
pmat1ovscd.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
pmat1ovscd.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
pmat1ovscd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑁) |
pmat1ovscd.j | ⊢ (𝜑 → 𝐽 ∈ 𝑁) |
pmat1ovscd.u | ⊢ 𝑈 = (1r‘𝐶) |
Ref | Expression |
---|---|
pmat1ovscd | ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, (𝐴‘ 1 ), (𝐴‘ 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmat0opsc.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pmat0opsc.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | eqid 2778 | . . 3 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | eqid 2778 | . . 3 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
5 | pmat1ovscd.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
6 | pmat1ovscd.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | pmat1ovscd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
8 | pmat1ovscd.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑁) | |
9 | pmat1ovscd.u | . . 3 ⊢ 𝑈 = (1r‘𝐶) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | pmat1ovd 20909 | . 2 ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, (1r‘𝑃), (0g‘𝑃))) |
11 | pmat0opsc.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑃) | |
12 | pmat1opsc.o | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
13 | 1, 11, 12, 4 | ply1scl1 20058 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 1 ) = (1r‘𝑃)) |
14 | 6, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 1 ) = (1r‘𝑃)) |
15 | 14 | eqcomd 2784 | . . 3 ⊢ (𝜑 → (1r‘𝑃) = (𝐴‘ 1 )) |
16 | pmat0opsc.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
17 | 1, 11, 16, 3 | ply1scl0 20056 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0g‘𝑃)) |
18 | 6, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0g‘𝑃)) |
19 | 18 | eqcomd 2784 | . . 3 ⊢ (𝜑 → (0g‘𝑃) = (𝐴‘ 0 )) |
20 | 15, 19 | ifeq12d 4327 | . 2 ⊢ (𝜑 → if(𝐼 = 𝐽, (1r‘𝑃), (0g‘𝑃)) = if(𝐼 = 𝐽, (𝐴‘ 1 ), (𝐴‘ 0 ))) |
21 | 10, 20 | eqtrd 2814 | 1 ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, (𝐴‘ 1 ), (𝐴‘ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ifcif 4307 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 0gc0g 16486 1rcur 18888 Ringcrg 18934 algSccascl 19708 Poly1cpl1 19943 Mat cmat 20617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-subrg 19170 df-lmod 19257 df-lss 19325 df-sra 19569 df-rgmod 19570 df-ascl 19711 df-psr 19753 df-mpl 19755 df-opsr 19757 df-psr1 19946 df-ply1 19948 df-dsmm 20475 df-frlm 20490 df-mamu 20594 df-mat 20618 |
This theorem is referenced by: 1elcpmat 20927 |
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