Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > marepvcl | Structured version Visualization version GIF version |
Description: Closure of the column replacement function for square matrices. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
Ref | Expression |
---|---|
marepvcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
marepvcl.b | ⊢ 𝐵 = (Base‘𝐴) |
marepvcl.v | ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) |
Ref | Expression |
---|---|
marepvcl | ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marepvcl.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | marepvcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
3 | eqid 2610 | . . . 4 ⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅) | |
4 | marepvcl.v | . . . 4 ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) | |
5 | 1, 2, 3, 4 | marepvval 20192 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)))) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)))) |
7 | eqid 2610 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | 1, 2 | matrcl 20037 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
9 | 8 | simpld 474 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
10 | 9 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin) |
11 | 10 | adantl 481 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝑁 ∈ Fin) |
12 | simpl 472 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
13 | elmapi 7765 | . . . . . . . . . 10 ⊢ (𝐶 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → 𝐶:𝑁⟶(Base‘𝑅)) | |
14 | ffvelrn 6265 | . . . . . . . . . . 11 ⊢ ((𝐶:𝑁⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝑁) → (𝐶‘𝑖) ∈ (Base‘𝑅)) | |
15 | 14 | ex 449 | . . . . . . . . . 10 ⊢ (𝐶:𝑁⟶(Base‘𝑅) → (𝑖 ∈ 𝑁 → (𝐶‘𝑖) ∈ (Base‘𝑅))) |
16 | 13, 15 | syl 17 | . . . . . . . . 9 ⊢ (𝐶 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → (𝑖 ∈ 𝑁 → (𝐶‘𝑖) ∈ (Base‘𝑅))) |
17 | 16, 4 | eleq2s 2706 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝑉 → (𝑖 ∈ 𝑁 → (𝐶‘𝑖) ∈ (Base‘𝑅))) |
18 | 17 | 3ad2ant2 1076 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → (𝑖 ∈ 𝑁 → (𝐶‘𝑖) ∈ (Base‘𝑅))) |
19 | 18 | adantl 481 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → (𝑖 ∈ 𝑁 → (𝐶‘𝑖) ∈ (Base‘𝑅))) |
20 | 19 | imp 444 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁) → (𝐶‘𝑖) ∈ (Base‘𝑅)) |
21 | 20 | 3adant3 1074 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝐶‘𝑖) ∈ (Base‘𝑅)) |
22 | simp2 1055 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁) | |
23 | simp3 1056 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) | |
24 | 2 | eleq2i 2680 | . . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴)) |
25 | 24 | biimpi 205 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴)) |
26 | 25 | 3ad2ant1 1075 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴)) |
27 | 26 | adantl 481 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐴)) |
28 | 27 | 3ad2ant1 1075 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴)) |
29 | 1, 7 | matecl 20050 | . . . . 5 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑖𝑀𝑗) ∈ (Base‘𝑅)) |
30 | 22, 23, 28, 29 | syl3anc 1318 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅)) |
31 | 21, 30 | ifcld 4081 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)) ∈ (Base‘𝑅)) |
32 | 1, 7, 2, 11, 12, 31 | matbas2d 20048 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))) ∈ 𝐵) |
33 | 6, 32 | eqeltrd 2688 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑𝑚 cmap 7744 Fincfn 7841 Basecbs 15695 Ringcrg 18370 Mat cmat 20032 matRepV cmatrepV 20182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-pws 15933 df-sra 18993 df-rgmod 18994 df-dsmm 19895 df-frlm 19910 df-mat 20033 df-marepv 20184 |
This theorem is referenced by: ma1repvcl 20195 |
Copyright terms: Public domain | W3C validator |