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Mirrors > Home > MPE Home > Th. List > mattposvs | Structured version Visualization version GIF version |
Description: The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Ref | Expression |
---|---|
mattposvs.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mattposvs.b | ⊢ 𝐵 = (Base‘𝐴) |
mattposvs.k | ⊢ 𝐾 = (Base‘𝑅) |
mattposvs.v | ⊢ · = ( ·𝑠 ‘𝐴) |
Ref | Expression |
---|---|
mattposvs | ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mattposvs.a | . . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | mattposvs.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐴) | |
3 | 1, 2 | matrcl 21281 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | 3 | simpld 498 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → 𝑁 ∈ Fin) |
5 | sqxpexg 7529 | . . . . . . 7 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ V) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (𝑁 × 𝑁) ∈ V) |
7 | snex 5313 | . . . . . 6 ⊢ {𝑋} ∈ V | |
8 | xpexg 7524 | . . . . . 6 ⊢ (((𝑁 × 𝑁) ∈ V ∧ {𝑋} ∈ V) → ((𝑁 × 𝑁) × {𝑋}) ∈ V) | |
9 | 6, 7, 8 | sylancl 589 | . . . . 5 ⊢ (𝑌 ∈ 𝐵 → ((𝑁 × 𝑁) × {𝑋}) ∈ V) |
10 | oftpos 21321 | . . . . 5 ⊢ ((((𝑁 × 𝑁) × {𝑋}) ∈ V ∧ 𝑌 ∈ 𝐵) → tpos (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)𝑌) = (tpos ((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌)) | |
11 | 9, 10 | mpancom 688 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → tpos (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)𝑌) = (tpos ((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌)) |
12 | tposconst 7995 | . . . . 5 ⊢ tpos ((𝑁 × 𝑁) × {𝑋}) = ((𝑁 × 𝑁) × {𝑋}) | |
13 | 12 | oveq1i 7212 | . . . 4 ⊢ (tpos ((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌) |
14 | 11, 13 | eqtrdi 2790 | . . 3 ⊢ (𝑌 ∈ 𝐵 → tpos (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌)) |
15 | 14 | adantl 485 | . 2 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → tpos (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌)) |
16 | mattposvs.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
17 | mattposvs.v | . . . 4 ⊢ · = ( ·𝑠 ‘𝐴) | |
18 | eqid 2734 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | eqid 2734 | . . . 4 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁) | |
20 | 1, 2, 16, 17, 18, 19 | matvsca2 21297 | . . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)𝑌)) |
21 | 20 | tposeqd 7960 | . 2 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = tpos (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)𝑌)) |
22 | 1, 2 | mattposcl 21322 | . . 3 ⊢ (𝑌 ∈ 𝐵 → tpos 𝑌 ∈ 𝐵) |
23 | 1, 2, 16, 17, 18, 19 | matvsca2 21297 | . . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ tpos 𝑌 ∈ 𝐵) → (𝑋 · tpos 𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌)) |
24 | 22, 23 | sylan2 596 | . 2 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · tpos 𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘f (.r‘𝑅)tpos 𝑌)) |
25 | 15, 21, 24 | 3eqtr4d 2784 | 1 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 {csn 4531 × cxp 5538 ‘cfv 6369 (class class class)co 7202 ∘f cof 7456 tpos ctpos 7956 Fincfn 8615 Basecbs 16684 .rcmulr 16768 ·𝑠 cvsca 16771 Mat cmat 21276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-ot 4540 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-prds 16924 df-pws 16926 df-sra 20181 df-rgmod 20182 df-dsmm 20666 df-frlm 20681 df-mat 21277 |
This theorem is referenced by: madulid 21514 |
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