Theorem xpcom 4992

Description: Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)

Assertion

Ref	Expression
xpcom	⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem xpcom

Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ibar 296	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶))))
2		ancom 263	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴))
3	2	anbi1i 447	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
4		brxp 4484	. . . . . . . 8 ⊢ (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
5		brxp 4484	. . . . . . . 8 ⊢ (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))
6	4, 5	anbi12i 449	. . . . . . 7 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
7		anandi 558	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
8	3, 6, 7	3bitr4i 211	. . . . . 6 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
9	8	exbii 1542	. . . . 5 ⊢ (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
10		19.41v 1831	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
11	9, 10	bitr2i 184	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐))
12	1, 11	syl6rbb 196	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
13	12	opabbidv 3912	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)})
14		df-co 4463	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)}
15		df-xp 4460	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)}
16	13, 14, 15	3eqtr4g 2146	1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290  ∃wex 1427   ∈ wcel 1439   class class class wbr 3853  {copab 3906   × cxp 4452   ∘ ccom 4458

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-co 4463

This theorem is referenced by: (None)