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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 2731 | . . 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 22208 | . 2 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))))) |
11 | ringcmn 20177 | . . . . . . 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 723 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
16 | elmapi 8849 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)) → 𝑋:(𝑀 × 𝑁)⟶𝐵) | |
17 | 8, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(𝑀 × 𝑁)⟶𝐵) |
18 | 17 | ad2antrr 723 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑋:(𝑀 × 𝑁)⟶𝐵) |
19 | simplrl 774 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑀) | |
20 | simpr 484 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) | |
21 | 18, 19, 20 | fovcdmd 7583 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵) |
22 | elmapi 8849 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)) → 𝑌:(𝑁 × 𝑃)⟶𝐵) | |
23 | 9, 22 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌:(𝑁 × 𝑃)⟶𝐵) |
24 | 23 | ad2antrr 723 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑌:(𝑁 × 𝑃)⟶𝐵) |
25 | simplrr 775 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ 𝑃) | |
26 | 24, 20, 25 | fovcdmd 7583 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → (𝑗𝑌𝑘) ∈ 𝐵) |
27 | 2, 3 | ringcl 20151 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑗𝑌𝑘) ∈ 𝐵) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
28 | 15, 21, 26, 27 | syl3anc 1370 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
29 | 28 | ralrimiva 3145 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
30 | 2, 13, 14, 29 | gsummptcl 19883 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑃)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) ∈ 𝐵) |
31 | 30 | ralrimivva 3199 | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑃 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) ∈ 𝐵) |
32 | eqid 2731 | . . . . 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))))) | |
33 | 32 | fmpo 8058 | . . . 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) |
34 | 2 | fvexi 6905 | . . . . 5 ⊢ 𝐵 ∈ V |
35 | xpfi 9323 | . . . . . 6 ⊢ ((𝑀 ∈ Fin ∧ 𝑃 ∈ Fin) → (𝑀 × 𝑃) ∈ Fin) | |
36 | 5, 7, 35 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑀 × 𝑃) ∈ Fin) |
37 | elmapg 8839 | . . . . 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)) | |
38 | 34, 36, 37 | sylancr 586 | . . . 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)) |
39 | 33, 38 | bitr4id 290 | . . 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)))) |
40 | 31, 39 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) ∈ (𝐵 ↑m (𝑀 × 𝑃))) |
41 | 10, 40 | eqeltrd 2832 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 〈cotp 4636 ↦ cmpt 5231 × cxp 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ↑m cmap 8826 Fincfn 8945 Basecbs 17151 .rcmulr 17205 Σg cgsu 17393 CMndccmn 19696 Ringcrg 20134 maMul cmmul 22205 |
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 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-0g 17394 df-gsum 17395 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-ur 20083 df-ring 20136 df-mamu 22206 |
This theorem is referenced by: mamuass 22222 mamudi 22223 mamudir 22224 mamuvs1 22225 mamuvs2 22226 mamulid 22263 mamurid 22264 matring 22265 matassa 22266 mavmulass 22371 |
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