Theorem xpcom 5329

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 301	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶))))
2		ancom 266	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴))
3	2	anbi1i 462	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
4		brxp 4800	. . . . . . . 8 ⊢ (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
5		brxp 4800	. . . . . . . 8 ⊢ (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))
6	4, 5	anbi12i 464	. . . . . . 7 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
7		anandi 598	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
8	3, 6, 7	3bitr4i 212	. . . . . 6 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
9	8	exbii 1658	. . . . 5 ⊢ (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
10		19.41v 1958	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
11	9, 10	bitr2i 185	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐))
12	1, 11	bitr2di 197	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
13	12	opabbidv 4192	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)})
14		df-co 4778	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)}
15		df-xp 4775	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)}
16	13, 14, 15	3eqtr4g 2296	1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209   class class class wbr 4125  {copab 4186   × cxp 4767   ∘ ccom 4773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-co 4778

This theorem is referenced by: (None)