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Theorem xpcom 4807
 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.
StepHypRef 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))
32anbi1i 431 . . . . . . 7 (((𝑎 A x B) (x B 𝑐 𝐶)) ↔ ((x B 𝑎 A) (x B 𝑐 𝐶)))
4 brxp 4318 . . . . . . . 8 (𝑎(A × B)x ↔ (𝑎 A x B))
5 brxp 4318 . . . . . . . 8 (x(B × 𝐶)𝑐 ↔ (x B 𝑐 𝐶))
64, 5anbi12i 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)))
83, 6, 73bitr4i 201 . . . . . 6 ((𝑎(A × B)x x(B × 𝐶)𝑐) ↔ (x B (𝑎 A 𝑐 𝐶)))
98exbii 1493 . . . . 5 (x(𝑎(A × B)x x(B × 𝐶)𝑐) ↔ x(x B (𝑎 A 𝑐 𝐶)))
10 19.41v 1779 . . . . 5 (x(x B (𝑎 A 𝑐 𝐶)) ↔ (x x B (𝑎 A 𝑐 𝐶)))
119, 10bitr2i 174 . . . 4 ((x x B (𝑎 A 𝑐 𝐶)) ↔ x(𝑎(A × B)x x(B × 𝐶)𝑐))
121, 11syl6rbb 186 . . 3 (x x B → (x(𝑎(A × B)x x(B × 𝐶)𝑐) ↔ (𝑎 A 𝑐 𝐶)))
1312opabbidv 3814 . 2 (x x B → {⟨𝑎, 𝑐⟩ ∣ x(𝑎(A × B)x x(B × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎 A 𝑐 𝐶)})
14 df-co 4297 . 2 ((B × 𝐶) ∘ (A × B)) = {⟨𝑎, 𝑐⟩ ∣ x(𝑎(A × B)x x(B × 𝐶)𝑐)}
15 df-xp 4294 . 2 (A × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎 A 𝑐 𝐶)}
1613, 14, 153eqtr4g 2094 1 (x x B → ((B × 𝐶) ∘ (A × B)) = (A × 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390   class class class wbr 3755  {copab 3808   × cxp 4286   ∘ ccom 4292 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-co 4297 This theorem is referenced by: (None)
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