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Mirrors > Home > MPE Home > Th. List > mattposcl | Structured version Visualization version GIF version |
Description: The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
mattposcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mattposcl.b | ⊢ 𝐵 = (Base‘𝐴) |
Ref | Expression |
---|---|
mattposcl | ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mattposcl.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | mattposcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
4 | 1, 2, 3 | matbas2i 20228 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
5 | elmapi 7879 | . . . 4 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) | |
6 | tposf 7380 | . . . 4 ⊢ (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅) → tpos 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
8 | fvex 6201 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
9 | 1, 3 | matrcl 20218 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
10 | 9 | simpld 475 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
11 | xpfi 8231 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) | |
12 | 11 | anidms 677 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
13 | 10, 12 | syl 17 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝑁 × 𝑁) ∈ Fin) |
14 | elmapg 7870 | . . . 4 ⊢ (((Base‘𝑅) ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → (tpos 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) ↔ tpos 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))) | |
15 | 8, 13, 14 | sylancr 695 | . . 3 ⊢ (𝑀 ∈ 𝐵 → (tpos 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) ↔ tpos 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))) |
16 | 7, 15 | mpbird 247 | . 2 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
17 | 1, 2 | matbas2 20227 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
18 | 9, 17 | syl 17 | . . 3 ⊢ (𝑀 ∈ 𝐵 → ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
19 | 18, 3 | syl6eqr 2674 | . 2 ⊢ (𝑀 ∈ 𝐵 → ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) = 𝐵) |
20 | 16, 19 | eleqtrd 2703 | 1 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 tpos ctpos 7351 ↑𝑚 cmap 7857 Fincfn 7955 Basecbs 15857 Mat cmat 20213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-pws 16110 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mat 20214 |
This theorem is referenced by: mattposvs 20261 mdettpos 20417 madutpos 20448 madulid 20451 mdetpmtr2 29890 |
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