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Mirrors > Home > MPE Home > Th. List > m2pmfzmap | Structured version Visualization version GIF version |
Description: The transformed values of a (finite) mapping of integers to matrices. (Contributed by AV, 4-Nov-2019.) |
Ref | Expression |
---|---|
m2pmfzmap.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
m2pmfzmap.b | ⊢ 𝐵 = (Base‘𝐴) |
m2pmfzmap.p | ⊢ 𝑃 = (Poly1‘𝑅) |
m2pmfzmap.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
m2pmfzmap.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
Ref | Expression |
---|---|
m2pmfzmap | ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → (𝑇‘(𝑏‘𝐼)) ∈ (Base‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1190 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → 𝑁 ∈ Fin) | |
2 | simpl2 1191 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → 𝑅 ∈ Ring) | |
3 | elmapi 8708 | . . . 4 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑆)) → 𝑏:(0...𝑆)⟶𝐵) | |
4 | 3 | ffvelcdmda 7017 | . . 3 ⊢ ((𝑏 ∈ (𝐵 ↑m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆)) → (𝑏‘𝐼) ∈ 𝐵) |
5 | 4 | adantl 482 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → (𝑏‘𝐼) ∈ 𝐵) |
6 | m2pmfzmap.t | . . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
7 | m2pmfzmap.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
8 | m2pmfzmap.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
9 | m2pmfzmap.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
10 | m2pmfzmap.y | . . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
11 | 6, 7, 8, 9, 10 | mat2pmatbas 21981 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝐼) ∈ 𝐵) → (𝑇‘(𝑏‘𝐼)) ∈ (Base‘𝑌)) |
12 | 1, 2, 5, 11 | syl3anc 1370 | 1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → (𝑇‘(𝑏‘𝐼)) ∈ (Base‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ‘cfv 6479 (class class class)co 7337 ↑m cmap 8686 Fincfn 8804 0cc0 10972 ℕ0cn0 12334 ...cfz 13340 Basecbs 17009 Ringcrg 19878 Poly1cpl1 21454 Mat cmat 21660 matToPolyMat cmat2pmat 21959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-ot 4582 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-ofr 7596 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-pm 8689 df-ixp 8757 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-sup 9299 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-fz 13341 df-fzo 13484 df-seq 13823 df-hash 14146 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-hom 17083 df-cco 17084 df-0g 17249 df-gsum 17250 df-prds 17255 df-pws 17257 df-mre 17392 df-mrc 17393 df-acs 17395 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-mhm 18527 df-submnd 18528 df-grp 18676 df-minusg 18677 df-sbg 18678 df-mulg 18797 df-subg 18848 df-ghm 18928 df-cntz 19019 df-cmn 19483 df-abl 19484 df-mgp 19816 df-ur 19833 df-ring 19880 df-subrg 20127 df-lmod 20231 df-lss 20300 df-sra 20540 df-rgmod 20541 df-dsmm 21045 df-frlm 21060 df-ascl 21168 df-psr 21218 df-mpl 21220 df-opsr 21222 df-psr1 21457 df-ply1 21459 df-mat 21661 df-mat2pmat 21962 |
This theorem is referenced by: m2pmfzgsumcl 22003 chfacfisf 22109 chfacfpmmulgsum2 22120 cpmadugsumlemB 22129 cpmadugsumlemC 22130 cpmadugsumlemF 22131 |
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