Theorem pprodss4v 36078

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.

Step	Hyp	Ref	Expression
1		df-pprod 36049	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		txprel 36073	. . 3 ⊢ Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3		txpss3v 36072	. . . . . . 7 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
4	3	sseli 3930	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5		opelxp2 5668	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) → 𝑦 ∈ (V × V))
6	4, 5	syl 17	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑦 ∈ (V × V))
7		elvv 5700	. . . . . 6 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8		opeq2 4831	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
9	8	eleq1d 2822	. . . . . . . 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 5100	. . . . . . . . 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
14	11, 12, 13	brtxp 36074	. . . . . . . . . 10 ⊢ (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ ↔ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ∧ 𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤))
15	11, 12	brco 5820	. . . . . . . . . . . 12 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧))
16		vex 3445	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
17	16	brresi 5948	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ↾ (V × V))𝑦 ↔ (𝑥 ∈ (V × V) ∧ 𝑥1st 𝑦))
18	17	simplbi 497	. . . . . . . . . . . . . 14 ⊢ (𝑥(1st ↾ (V × V))𝑦 → 𝑥 ∈ (V × V))
19	18	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
20	19	exlimiv 1932	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
21	15, 20	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 → 𝑥 ∈ (V × V))
22	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ∧ 𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤) → 𝑥 ∈ (V × V))
23	14, 22	sylbi 217	. . . . . . . . 9 ⊢ (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ → 𝑥 ∈ (V × V))
24	10, 23	sylbir 235	. . . . . . . 8 ⊢ (⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
25	9, 24	biimtrdi 253	. . . . . . 7 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
26	25	exlimivv 1934	. . . . . 6 ⊢ (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
27	7, 26	sylbi 217	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
28	6, 27	mpcom 38	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
29	28, 6	opelxpd 5664	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
30	2, 29	relssi 5737	. 2 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
31	1, 30	eqsstri 3981	1 ⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3441   ⊆ wss 3902  ⟨cop 4587   class class class wbr 5099   × cxp 5623   ↾ cres 5627   ∘ ccom 5629  1^st c1st 7933  2^nd c2nd 7934   ⊗ ctxp 36024  pprodcpprod 36025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7935  df-2nd 7936  df-txp 36048  df-pprod 36049

This theorem is referenced by:  brpprod3a  36080