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Theorem pprodss4v 32116
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.)
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 32087 . 2 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2 txprel 32111 . . 3 Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3 txpss3v 32110 . . . . . . 7 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
43sseli 3632 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5 opelxp2 5185 . . . . . 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 5211 . . . . . 6 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8 opeq2 4434 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
98eleq1d 2715 . . . . . . . 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 4686 . . . . . . . . 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 3234 . . . . . . . . . . 11 𝑥 ∈ V
12 vex 3234 . . . . . . . . . . 11 𝑧 ∈ V
13 vex 3234 . . . . . . . . . . 11 𝑤 ∈ V
1411, 12, 13brtxp 32112 . . . . . . . . . 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 5325 . . . . . . . . . . . 12 (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧))
16 vex 3234 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1716brres 5437 . . . . . . . . . . . . . . 15 (𝑥(1st ↾ (V × V))𝑦 ↔ (𝑥1st 𝑦𝑥 ∈ (V × V)))
1817simprbi 479 . . . . . . . . . . . . . 14 (𝑥(1st ↾ (V × V))𝑦𝑥 ∈ (V × V))
1918adantr 480 . . . . . . . . . . . . 13 ((𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
2019exlimiv 1898 . . . . . . . . . . . 12 (∃𝑦(𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
2115, 20sylbi 207 . . . . . . . . . . 11 (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥 ∈ (V × V))
2221adantr 480 . . . . . . . . . 10 ((𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤) → 𝑥 ∈ (V × V))
2314, 22sylbi 207 . . . . . . . . 9 (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ → 𝑥 ∈ (V × V))
2410, 23sylbir 225 . . . . . . . 8 (⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
259, 24syl6bi 243 . . . . . . 7 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
2625exlimivv 1900 . . . . . 6 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
277, 26sylbi 207 . . . . 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))
29 opelxp 5180 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)) ↔ (𝑥 ∈ (V × V) ∧ 𝑦 ∈ (V × V)))
3028, 6, 29sylanbrc 699 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
312, 30relssi 5245 . 2 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
321, 31eqsstri 3668 1 pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  wss 3607  cop 4216   class class class wbr 4685   × cxp 5141  cres 5145  ccom 5147  1st c1st 7208  2nd c2nd 7209  ctxp 32062  pprodcpprod 32063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-pprod 32087
This theorem is referenced by:  brpprod3a  32118
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