Theorem xpcom 4864

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 285	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶))))
2		ancom 253	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴))
3	2	anbi1i 431	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
4		brxp 4375	. . . . . . . 8 ⊢ (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
5		brxp 4375	. . . . . . . 8 ⊢ (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))
6	4, 5	anbi12i 433	. . . . . . 7 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
7		anandi 524	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
8	3, 6, 7	3bitr4i 201	. . . . . 6 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
9	8	exbii 1496	. . . . 5 ⊢ (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
10		19.41v 1782	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
11	9, 10	bitr2i 174	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐))
12	1, 11	syl6rbb 186	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
13	12	opabbidv 3823	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)})
14		df-co 4354	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)}
15		df-xp 4351	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)}
16	13, 14, 15	3eqtr4g 2097	1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   class class class wbr 3764  {copab 3817   × cxp 4343   ∘ ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-co 4354

This theorem is referenced by: (None)