MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposmap Structured version   Visualization version   GIF version

Theorem tposmap 20540
Description: The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
Assertion
Ref Expression
tposmap (𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵𝑚 (𝐽 × 𝐼)))

Proof of Theorem tposmap
StepHypRef Expression
1 elmapi 8082 . . 3 (𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → 𝐴:(𝐼 × 𝐽)⟶𝐵)
2 tposf 7583 . . 3 (𝐴:(𝐼 × 𝐽)⟶𝐵 → tpos 𝐴:(𝐽 × 𝐼)⟶𝐵)
31, 2syl 17 . 2 (𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → tpos 𝐴:(𝐽 × 𝐼)⟶𝐵)
4 elmapex 8081 . . 3 (𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → (𝐵 ∈ V ∧ (𝐼 × 𝐽) ∈ V))
5 cnvxp 5734 . . . . 5 (𝐼 × 𝐽) = (𝐽 × 𝐼)
6 cnvexg 7310 . . . . 5 ((𝐼 × 𝐽) ∈ V → (𝐼 × 𝐽) ∈ V)
75, 6syl5eqelr 2849 . . . 4 ((𝐼 × 𝐽) ∈ V → (𝐽 × 𝐼) ∈ V)
87anim2i 610 . . 3 ((𝐵 ∈ V ∧ (𝐼 × 𝐽) ∈ V) → (𝐵 ∈ V ∧ (𝐽 × 𝐼) ∈ V))
9 elmapg 8073 . . 3 ((𝐵 ∈ V ∧ (𝐽 × 𝐼) ∈ V) → (tpos 𝐴 ∈ (𝐵𝑚 (𝐽 × 𝐼)) ↔ tpos 𝐴:(𝐽 × 𝐼)⟶𝐵))
104, 8, 93syl 18 . 2 (𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → (tpos 𝐴 ∈ (𝐵𝑚 (𝐽 × 𝐼)) ↔ tpos 𝐴:(𝐽 × 𝐼)⟶𝐵))
113, 10mpbird 248 1 (𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵𝑚 (𝐽 × 𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2155  Vcvv 3350   × cxp 5275  ccnv 5276  wf 6064  (class class class)co 6842  tpos ctpos 7554  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-tpos 7555  df-map 8062
This theorem is referenced by:  mamutpos  20541
  Copyright terms: Public domain W3C validator