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Theorem pprodss4v 34277
Description: The parallel product is a subclass of ((V × V) × (V × V)). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
pprodss4v pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Proof of Theorem pprodss4v
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pprod 34248 . 2 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2 txprel 34272 . . 3 Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3 txpss3v 34271 . . . . . . 7 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
43sseli 3927 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5 opelxp2 5656 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) → 𝑦 ∈ (V × V))
64, 5syl 17 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑦 ∈ (V × V))
7 elvv 5686 . . . . . 6 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8 opeq2 4817 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
98eleq1d 2821 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ↔ ⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))))
10 df-br 5090 . . . . . . . . 9 (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ ↔ ⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))))
11 vex 3445 . . . . . . . . . . 11 𝑥 ∈ V
12 vex 3445 . . . . . . . . . . 11 𝑧 ∈ V
13 vex 3445 . . . . . . . . . . 11 𝑤 ∈ V
1411, 12, 13brtxp 34273 . . . . . . . . . 10 (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ ↔ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤))
1511, 12brco 5806 . . . . . . . . . . . 12 (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧))
16 vex 3445 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1716brresi 5926 . . . . . . . . . . . . . . 15 (𝑥(1st ↾ (V × V))𝑦 ↔ (𝑥 ∈ (V × V) ∧ 𝑥1st 𝑦))
1817simplbi 498 . . . . . . . . . . . . . 14 (𝑥(1st ↾ (V × V))𝑦𝑥 ∈ (V × V))
1918adantr 481 . . . . . . . . . . . . 13 ((𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
2019exlimiv 1932 . . . . . . . . . . . 12 (∃𝑦(𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
2115, 20sylbi 216 . . . . . . . . . . 11 (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥 ∈ (V × V))
2221adantr 481 . . . . . . . . . 10 ((𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤) → 𝑥 ∈ (V × V))
2314, 22sylbi 216 . . . . . . . . 9 (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ → 𝑥 ∈ (V × V))
2410, 23sylbir 234 . . . . . . . 8 (⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
259, 24syl6bi 252 . . . . . . 7 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
2625exlimivv 1934 . . . . . 6 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
277, 26sylbi 216 . . . . 5 (𝑦 ∈ (V × V) → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
286, 27mpcom 38 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
2928, 6opelxpd 5652 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
302, 29relssi 5723 . 2 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
311, 30eqsstri 3965 1 pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wex 1780  wcel 2105  Vcvv 3441  wss 3897  cop 4578   class class class wbr 5089   × cxp 5612  cres 5616  ccom 5618  1st c1st 7889  2nd c2nd 7890  ctxp 34223  pprodcpprod 34224
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-fo 6479  df-fv 6481  df-1st 7891  df-2nd 7892  df-txp 34247  df-pprod 34248
This theorem is referenced by:  brpprod3a  34279
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