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Theorem tposmap 21042
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 8406 . . 3 (𝐴 ∈ (𝐵m (𝐼 × 𝐽)) → 𝐴:(𝐼 × 𝐽)⟶𝐵)
2 tposf 7898 . . 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 8405 . . 3 (𝐴 ∈ (𝐵m (𝐼 × 𝐽)) → (𝐵 ∈ V ∧ (𝐼 × 𝐽) ∈ V))
5 cnvxp 5990 . . . . 5 (𝐼 × 𝐽) = (𝐽 × 𝐼)
6 cnvexg 7607 . . . . 5 ((𝐼 × 𝐽) ∈ V → (𝐼 × 𝐽) ∈ V)
75, 6eqeltrrid 2916 . . . 4 ((𝐼 × 𝐽) ∈ V → (𝐽 × 𝐼) ∈ V)
87anim2i 618 . . 3 ((𝐵 ∈ V ∧ (𝐼 × 𝐽) ∈ V) → (𝐵 ∈ V ∧ (𝐽 × 𝐼) ∈ V))
9 elmapg 8397 . . 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 259 1 (𝐴 ∈ (𝐵m (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵m (𝐽 × 𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2114  Vcvv 3473   × cxp 5529  ccnv 5530  wf 6327  (class class class)co 7133  tpos ctpos 7869  m cmap 8384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fo 6337  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-tpos 7870  df-map 8386
This theorem is referenced by:  mamutpos  21043
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