Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mavmul0 | Structured version Visualization version GIF version |
Description: Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.) |
Ref | Expression |
---|---|
mavmul0.t | ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) |
Ref | Expression |
---|---|
mavmul0 | ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
2 | mavmul0.t | . . 3 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
3 | eqid 2824 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2824 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | simpr 487 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
6 | 0fin 8749 | . . . . 5 ⊢ ∅ ∈ Fin | |
7 | eleq1 2903 | . . . . 5 ⊢ (𝑁 = ∅ → (𝑁 ∈ Fin ↔ ∅ ∈ Fin)) | |
8 | 6, 7 | mpbiri 260 | . . . 4 ⊢ (𝑁 = ∅ → 𝑁 ∈ Fin) |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin) |
10 | 0ex 5214 | . . . . 5 ⊢ ∅ ∈ V | |
11 | snidg 4602 | . . . . 5 ⊢ (∅ ∈ V → ∅ ∈ {∅}) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ {∅}) |
13 | oveq1 7166 | . . . . . . 7 ⊢ (𝑁 = ∅ → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) | |
14 | 13 | adantr 483 | . . . . . 6 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) |
15 | 14 | fveq2d 6677 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(∅ Mat 𝑅))) |
16 | mat0dimbas0 21078 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) | |
17 | 16 | adantl 484 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(∅ Mat 𝑅)) = {∅}) |
18 | 15, 17 | eqtrd 2859 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = {∅}) |
19 | 12, 18 | eleqtrrd 2919 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ (Base‘(𝑁 Mat 𝑅))) |
20 | eqidd 2825 | . . . . . 6 ⊢ (𝑁 = ∅ → ∅ = ∅) | |
21 | el1o 8127 | . . . . . 6 ⊢ (∅ ∈ 1o ↔ ∅ = ∅) | |
22 | 20, 21 | sylibr 236 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ ∈ 1o) |
23 | oveq2 7167 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = ((Base‘𝑅) ↑m ∅)) | |
24 | fvex 6686 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
25 | map0e 8449 | . . . . . . 7 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑m ∅) = 1o) | |
26 | 24, 25 | mp1i 13 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m ∅) = 1o) |
27 | 23, 26 | eqtrd 2859 | . . . . 5 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = 1o) |
28 | 22, 27 | eleqtrrd 2919 | . . . 4 ⊢ (𝑁 = ∅ → ∅ ∈ ((Base‘𝑅) ↑m 𝑁)) |
29 | 28 | adantr 483 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ ((Base‘𝑅) ↑m 𝑁)) |
30 | 1, 2, 3, 4, 5, 9, 19, 29 | mavmulval 21157 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) |
31 | mpteq1 5157 | . . . 4 ⊢ (𝑁 = ∅ → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) | |
32 | 31 | adantr 483 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) |
33 | mpt0 6493 | . . 3 ⊢ (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅ | |
34 | 32, 33 | syl6eq 2875 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅) |
35 | 30, 34 | eqtrd 2859 | 1 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 {csn 4570 〈cop 4576 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 1oc1o 8098 ↑m cmap 8409 Fincfn 8512 Basecbs 16486 .rcmulr 16569 Σg cgsu 16717 Mat cmat 21019 maVecMul cmvmul 21152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-hom 16592 df-cco 16593 df-0g 16718 df-prds 16724 df-pws 16726 df-sra 19947 df-rgmod 19948 df-dsmm 20879 df-frlm 20894 df-mat 21020 df-mvmul 21153 |
This theorem is referenced by: mavmul0g 21165 cramer0 21302 |
Copyright terms: Public domain | W3C validator |