Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2737 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | mattposcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
4 | 1, 2, 3 | matbas2i 21676 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
5 | elmapi 8712 | . . . 4 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) | |
6 | tposf 8144 | . . . 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 6842 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
9 | 1, 3 | matrcl 21664 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
10 | 9 | simpld 496 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
11 | xpfi 9186 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) | |
12 | 11 | anidms 568 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
13 | 10, 12 | syl 17 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝑁 × 𝑁) ∈ Fin) |
14 | elmapg 8703 | . . . 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 588 | . . 3 ⊢ (𝑀 ∈ 𝐵 → (tpos 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) ↔ tpos 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))) |
16 | 7, 15 | mpbird 257 | . 2 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
17 | 1, 2 | matbas2 21675 | . . . 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 | eqtr4di 2795 | . 2 ⊢ (𝑀 ∈ 𝐵 → ((Base‘𝑅) ↑m (𝑁 × 𝑁)) = 𝐵) |
20 | 16, 19 | eleqtrd 2840 | 1 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1541 ∈ wcel 2106 Vcvv 3442 × cxp 5622 ⟶wf 6479 ‘cfv 6483 (class class class)co 7341 tpos ctpos 8115 ↑m cmap 8690 Fincfn 8808 Basecbs 17009 Mat cmat 21659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-resscn 11033 ax-1cn 11034 ax-icn 11035 ax-addcl 11036 ax-addrcl 11037 ax-mulcl 11038 ax-mulrcl 11039 ax-mulcom 11040 ax-addass 11041 ax-mulass 11042 ax-distr 11043 ax-i2m1 11044 ax-1ne0 11045 ax-1rid 11046 ax-rnegex 11047 ax-rrecex 11048 ax-cnre 11049 ax-pre-lttri 11050 ax-pre-lttrn 11051 ax-pre-ltadd 11052 ax-pre-mulgt0 11053 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4857 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-1st 7903 df-2nd 7904 df-supp 8052 df-tpos 8116 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-1o 8371 df-er 8573 df-map 8692 df-ixp 8761 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-fsupp 9231 df-sup 9303 df-pnf 11116 df-mnf 11117 df-xr 11118 df-ltxr 11119 df-le 11120 df-sub 11312 df-neg 11313 df-nn 12079 df-2 12141 df-3 12142 df-4 12143 df-5 12144 df-6 12145 df-7 12146 df-8 12147 df-9 12148 df-n0 12339 df-z 12425 df-dec 12543 df-uz 12688 df-fz 13345 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-hom 17083 df-cco 17084 df-0g 17249 df-prds 17255 df-pws 17257 df-sra 20539 df-rgmod 20540 df-dsmm 21044 df-frlm 21059 df-mat 21660 |
This theorem is referenced by: mattposvs 21709 mdettpos 21865 madutpos 21896 madulid 21899 mdetpmtr2 32070 |
Copyright terms: Public domain | W3C validator |