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Theorem dmatmulcl 20246
Description: The product of two diagonal matrices is a diagonal matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
dmatid.a 𝐴 = (𝑁 Mat 𝑅)
dmatid.b 𝐵 = (Base‘𝐴)
dmatid.0 0 = (0g𝑅)
dmatid.d 𝐷 = (𝑁 DMat 𝑅)
Assertion
Ref Expression
dmatmulcl (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) ∈ 𝐷)

Proof of Theorem dmatmulcl
Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmatid.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 eqid 2621 . . . 4 (Base‘𝑅) = (Base‘𝑅)
3 dmatid.b . . . 4 𝐵 = (Base‘𝐴)
4 simpll 789 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑁 ∈ Fin)
5 simplr 791 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ Ring)
653ad2ant1 1080 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Ring)
7 eqid 2621 . . . . . . 7 (Base‘𝐴) = (Base‘𝐴)
8 simp2 1060 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑥𝑁)
9 simp3 1061 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑦𝑁)
10 dmatid.0 . . . . . . . . . . 11 0 = (0g𝑅)
11 dmatid.d . . . . . . . . . . 11 𝐷 = (𝑁 DMat 𝑅)
121, 7, 10, 11dmatmat 20240 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷𝑋 ∈ (Base‘𝐴)))
1312imp 445 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋 ∈ (Base‘𝐴))
1413adantrr 752 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ (Base‘𝐴))
15143ad2ant1 1080 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑋 ∈ (Base‘𝐴))
161, 2, 7, 8, 9, 15matecld 20172 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
171, 7, 10, 11dmatmat 20240 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌𝐷𝑌 ∈ (Base‘𝐴)))
1817imp 445 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌 ∈ (Base‘𝐴))
1918adantrl 751 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ (Base‘𝐴))
20193ad2ant1 1080 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑌 ∈ (Base‘𝐴))
211, 2, 7, 8, 9, 20matecld 20172 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
22 eqid 2621 . . . . . . 7 (.r𝑅) = (.r𝑅)
232, 22ringcl 18501 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑦) ∈ (Base‘𝑅) ∧ (𝑥𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
246, 16, 21, 23syl3anc 1323 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
252, 10ring0cl 18509 . . . . . . . 8 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
2625adantl 482 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ (Base‘𝑅))
2726adantr 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 0 ∈ (Base‘𝑅))
28273ad2ant1 1080 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 0 ∈ (Base‘𝑅))
2924, 28ifcld 4109 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) ∈ (Base‘𝑅))
301, 2, 3, 4, 5, 29matbas2d 20169 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) ∈ 𝐵)
31 eqidd 2622 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) → (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )))
32 eqeq12 2634 . . . . . . . . 9 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥 = 𝑦𝑖 = 𝑗))
33 oveq12 6624 . . . . . . . . . 10 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥𝑋𝑦) = (𝑖𝑋𝑗))
34 oveq12 6624 . . . . . . . . . 10 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥𝑌𝑦) = (𝑖𝑌𝑗))
3533, 34oveq12d 6633 . . . . . . . . 9 ((𝑥 = 𝑖𝑦 = 𝑗) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) = ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)))
3632, 35ifbieq1d 4087 . . . . . . . 8 ((𝑥 = 𝑖𝑦 = 𝑗) → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = if(𝑖 = 𝑗, ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)), 0 ))
3736adantl 482 . . . . . . 7 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) ∧ (𝑥 = 𝑖𝑦 = 𝑗)) → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = if(𝑖 = 𝑗, ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)), 0 ))
38 simplrl 799 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) → 𝑖𝑁)
39 simplrr 800 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) → 𝑗𝑁)
40 ovex 6643 . . . . . . . . 9 ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)) ∈ V
41 fvex 6168 . . . . . . . . . 10 (0g𝑅) ∈ V
4210, 41eqeltri 2694 . . . . . . . . 9 0 ∈ V
4340, 42ifex 4134 . . . . . . . 8 if(𝑖 = 𝑗, ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)), 0 ) ∈ V
4443a1i 11 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) → if(𝑖 = 𝑗, ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)), 0 ) ∈ V)
4531, 37, 38, 39, 44ovmpt2d 6753 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = if(𝑖 = 𝑗, ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)), 0 ))
46 ifnefalse 4076 . . . . . . 7 (𝑖𝑗 → if(𝑖 = 𝑗, ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)), 0 ) = 0 )
4746adantl 482 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) → if(𝑖 = 𝑗, ((𝑖𝑋𝑗)(.r𝑅)(𝑖𝑌𝑗)), 0 ) = 0 )
4845, 47eqtrd 2655 . . . . 5 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑖𝑗) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = 0 )
4948ex 450 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖𝑗 → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = 0 ))
5049ralrimivva 2967 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = 0 ))
51 oveq 6621 . . . . . . 7 (𝑚 = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) → (𝑖𝑚𝑗) = (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗))
5251eqeq1d 2623 . . . . . 6 (𝑚 = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) → ((𝑖𝑚𝑗) = 0 ↔ (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = 0 ))
5352imbi2d 330 . . . . 5 (𝑚 = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) → ((𝑖𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ (𝑖𝑗 → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = 0 )))
54532ralbidv 2985 . . . 4 (𝑚 = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = 0 )))
5554elrab 3351 . . 3 ((𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ↔ ((𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) ∈ 𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))𝑗) = 0 )))
5630, 50, 55sylanbrc 697 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )) ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
571, 3, 10, 11dmatmul 20243 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )))
581, 3, 10, 11dmatval 20238 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
5958adantr 481 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
6056, 57, 593eltr4d 2713 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  {crab 2912  Vcvv 3190  ifcif 4064  cfv 5857  (class class class)co 6615  cmpt2 6617  Fincfn 7915  Basecbs 15800  .rcmulr 15882  0gc0g 16040  Ringcrg 18487   Mat cmat 20153   DMat cdmat 20234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-ot 4164  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-sup 8308  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-hom 15906  df-cco 15907  df-0g 16042  df-gsum 16043  df-prds 16048  df-pws 16050  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-grp 17365  df-minusg 17366  df-mulg 17481  df-cntz 17690  df-cmn 18135  df-abl 18136  df-mgp 18430  df-ur 18442  df-ring 18489  df-sra 19112  df-rgmod 19113  df-dsmm 20016  df-frlm 20031  df-mamu 20130  df-mat 20154  df-dmat 20236
This theorem is referenced by:  dmatsrng  20247
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