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Mirrors > Home > MPE Home > Th. List > tposmap | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
tposmap | ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵 ↑𝑚 (𝐽 × 𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 7765 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (𝐼 × 𝐽)) → 𝐴:(𝐼 × 𝐽)⟶𝐵) | |
2 | tposf 7267 | . . 3 ⊢ (𝐴:(𝐼 × 𝐽)⟶𝐵 → tpos 𝐴:(𝐽 × 𝐼)⟶𝐵) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (𝐼 × 𝐽)) → tpos 𝐴:(𝐽 × 𝐼)⟶𝐵) |
4 | elmapex 7764 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (𝐼 × 𝐽)) → (𝐵 ∈ V ∧ (𝐼 × 𝐽) ∈ V)) | |
5 | cnvxp 5470 | . . . . 5 ⊢ ◡(𝐼 × 𝐽) = (𝐽 × 𝐼) | |
6 | cnvexg 7005 | . . . . 5 ⊢ ((𝐼 × 𝐽) ∈ V → ◡(𝐼 × 𝐽) ∈ V) | |
7 | 5, 6 | syl5eqelr 2693 | . . . 4 ⊢ ((𝐼 × 𝐽) ∈ V → (𝐽 × 𝐼) ∈ V) |
8 | 7 | anim2i 591 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝐼 × 𝐽) ∈ V) → (𝐵 ∈ V ∧ (𝐽 × 𝐼) ∈ V)) |
9 | elmapg 7757 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝐽 × 𝐼) ∈ V) → (tpos 𝐴 ∈ (𝐵 ↑𝑚 (𝐽 × 𝐼)) ↔ tpos 𝐴:(𝐽 × 𝐼)⟶𝐵)) | |
10 | 4, 8, 9 | 3syl 18 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (𝐼 × 𝐽)) → (tpos 𝐴 ∈ (𝐵 ↑𝑚 (𝐽 × 𝐼)) ↔ tpos 𝐴:(𝐽 × 𝐼)⟶𝐵)) |
11 | 3, 10 | mpbird 246 | 1 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵 ↑𝑚 (𝐽 × 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 × cxp 5036 ◡ccnv 5037 ⟶wf 5800 (class class class)co 6549 tpos ctpos 7238 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-tpos 7239 df-map 7746 |
This theorem is referenced by: mamutpos 20083 |
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