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Mirrors > Home > MPE Home > Th. List > mattposm | Structured version Visualization version GIF version |
Description: Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Ref | Expression |
---|---|
mattposm.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mattposm.b | ⊢ 𝐵 = (Base‘𝐴) |
mattposm.t | ⊢ · = (.r‘𝐴) |
Ref | Expression |
---|---|
mattposm | ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
2 | eqid 2824 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | simp1 1132 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ CRing) | |
4 | mattposm.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | mattposm.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
6 | 4, 5 | matrcl 21024 | . . . . 5 ⊢ (𝑌 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
7 | 6 | simpld 497 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑁 ∈ Fin) |
8 | 7 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑁 ∈ Fin) |
9 | 4, 2, 5 | matbas2i 21034 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
10 | 9 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
11 | 4, 2, 5 | matbas2i 21034 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
12 | 11 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
13 | 1, 1, 2, 3, 8, 8, 8, 10, 12 | mamutpos 21070 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑌) = (tpos 𝑌(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)tpos 𝑋)) |
14 | 4, 1 | matmulr 21050 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
15 | 8, 3, 14 | syl2anc 586 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
16 | mattposm.t | . . . . 5 ⊢ · = (.r‘𝐴) | |
17 | 15, 16 | syl6reqr 2878 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)) |
18 | 17 | oveqd 7176 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑋(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑌)) |
19 | 18 | tposeqd 7898 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = tpos (𝑋(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑌)) |
20 | 17 | oveqd 7176 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (tpos 𝑌 · tpos 𝑋) = (tpos 𝑌(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)tpos 𝑋)) |
21 | 13, 19, 20 | 3eqtr4d 2869 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 〈cotp 4578 × cxp 5556 ‘cfv 6358 (class class class)co 7159 tpos ctpos 7894 ↑m cmap 8409 Fincfn 8512 Basecbs 16486 .rcmulr 16569 CRingccrg 19301 maMul cmmul 20997 Mat cmat 21019 |
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-tpos 7895 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-cmn 18911 df-mgp 19243 df-cring 19303 df-sra 19947 df-rgmod 19948 df-dsmm 20879 df-frlm 20894 df-mamu 20998 df-mat 21020 |
This theorem is referenced by: madulid 21257 |
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