Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mamucl | Structured version Visualization version GIF version |
Description: Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
Ref | Expression |
---|---|
mamucl.b | ⊢ 𝐵 = (Base‘𝑅) |
mamucl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mamucl.f | ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) |
mamucl.m | ⊢ (𝜑 → 𝑀 ∈ Fin) |
mamucl.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mamucl.p | ⊢ (𝜑 → 𝑃 ∈ Fin) |
mamucl.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) |
mamucl.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃))) |
Ref | Expression |
---|---|
mamucl | ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑𝑚 (𝑀 × 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mamucl.f | . . 3 ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) | |
2 | mamucl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | eqid 2610 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | mamucl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
5 | mamucl.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
6 | mamucl.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
7 | mamucl.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ Fin) | |
8 | mamucl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) | |
9 | mamucl.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mamuval 20011 | . 2 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))))) |
11 | ringcmn 18404 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | |
12 | 4, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) → 𝑅 ∈ CMnd) |
14 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) → 𝑁 ∈ Fin) |
15 | 4 | ad2antrr 758 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
16 | elmapi 7765 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)) → 𝑋:(𝑀 × 𝑁)⟶𝐵) | |
17 | 8, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(𝑀 × 𝑁)⟶𝐵) |
18 | 17 | ad2antrr 758 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑋:(𝑀 × 𝑁)⟶𝐵) |
19 | simplrl 796 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑀) | |
20 | simpr 476 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) | |
21 | 18, 19, 20 | fovrnd 6704 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵) |
22 | elmapi 7765 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃)) → 𝑌:(𝑁 × 𝑃)⟶𝐵) | |
23 | 9, 22 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌:(𝑁 × 𝑃)⟶𝐵) |
24 | 23 | ad2antrr 758 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑌:(𝑁 × 𝑃)⟶𝐵) |
25 | simplrr 797 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ 𝑃) | |
26 | 24, 20, 25 | fovrnd 6704 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → (𝑗𝑌𝑘) ∈ 𝐵) |
27 | 2, 3 | ringcl 18384 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑗𝑌𝑘) ∈ 𝐵) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
28 | 15, 21, 26, 27 | syl3anc 1318 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
29 | 28 | ralrimiva 2949 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
30 | 2, 13, 14, 29 | gsummptcl 18189 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) ∈ 𝐵) |
31 | 30 | ralrimivva 2954 | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑃 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) ∈ 𝐵) |
32 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
33 | 2, 32 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
34 | xpfi 8116 | . . . . . 6 ⊢ ((𝑀 ∈ Fin ∧ 𝑃 ∈ Fin) → (𝑀 × 𝑃) ∈ Fin) | |
35 | 5, 7, 34 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑀 × 𝑃) ∈ Fin) |
36 | elmapg 7757 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑃) ∈ Fin) → ((𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) ∈ (𝐵 ↑𝑚 (𝑀 × 𝑃)) ↔ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))):(𝑀 × 𝑃)⟶𝐵)) | |
37 | 33, 35, 36 | sylancr 694 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) ∈ (𝐵 ↑𝑚 (𝑀 × 𝑃)) ↔ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))):(𝑀 × 𝑃)⟶𝐵)) |
38 | eqid 2610 | . . . . 5 ⊢ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) | |
39 | 38 | fmpt2 7126 | . . . 4 ⊢ (∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑃 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))):(𝑀 × 𝑃)⟶𝐵) |
40 | 37, 39 | syl6rbbr 278 | . . 3 ⊢ (𝜑 → (∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑃 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) ∈ (𝐵 ↑𝑚 (𝑀 × 𝑃)))) |
41 | 31, 40 | mpbid 221 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) ∈ (𝐵 ↑𝑚 (𝑀 × 𝑃))) |
42 | 10, 41 | eqeltrd 2688 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑𝑚 (𝑀 × 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 〈cotp 4133 ↦ cmpt 4643 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑𝑚 cmap 7744 Fincfn 7841 Basecbs 15695 .rcmulr 15769 Σg cgsu 15924 CMndccmn 18016 Ringcrg 18370 maMul cmmul 20008 |
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-rmo 2904 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-se 4998 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-isom 5813 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-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 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-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-mamu 20009 |
This theorem is referenced by: mamuass 20027 mamudi 20028 mamudir 20029 mamuvs1 20030 mamuvs2 20031 mamulid 20066 mamurid 20067 matring 20068 matassa 20069 mavmulass 20174 |
Copyright terms: Public domain | W3C validator |