Theorem xpcom 4884

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 295	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶))))
2		ancom 262	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴))
3	2	anbi1i 445	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
4		brxp 4393	. . . . . . . 8 ⊢ (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
5		brxp 4393	. . . . . . . 8 ⊢ (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))
6	4, 5	anbi12i 447	. . . . . . 7 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
7		anandi 554	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
8	3, 6, 7	3bitr4i 210	. . . . . 6 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
9	8	exbii 1536	. . . . 5 ⊢ (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
10		19.41v 1823	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
11	9, 10	bitr2i 183	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐))
12	1, 11	syl6rbb 195	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
13	12	opabbidv 3844	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)})
14		df-co 4372	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)}
15		df-xp 4369	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)}
16	13, 14, 15	3eqtr4g 2138	1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433   class class class wbr 3785  {copab 3838   × cxp 4361   ∘ ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-co 4372

This theorem is referenced by: (None)