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Theorem restxp 5786
 Description: Restriction distributes over tail cross product. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
restxp ((AB) C) = ((A C) ⊗ (B C))

Proof of Theorem restxp
Dummy variables a b x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anandir 802 . . . . . 6 (((xAa xBb) x C) ↔ ((xAa x C) (xBb x C)))
21anbi2i 675 . . . . 5 ((y = a, b ((xAa xBb) x C)) ↔ (y = a, b ((xAa x C) (xBb x C))))
3 3anass 938 . . . . . . 7 ((y = a, b xAa xBb) ↔ (y = a, b (xAa xBb)))
43anbi1i 676 . . . . . 6 (((y = a, b xAa xBb) x C) ↔ ((y = a, b (xAa xBb)) x C))
5 anass 630 . . . . . 6 (((y = a, b (xAa xBb)) x C) ↔ (y = a, b ((xAa xBb) x C)))
64, 5bitri 240 . . . . 5 (((y = a, b xAa xBb) x C) ↔ (y = a, b ((xAa xBb) x C)))
7 3anass 938 . . . . 5 ((y = a, b (xAa x C) (xBb x C)) ↔ (y = a, b ((xAa x C) (xBb x C))))
82, 6, 73bitr4i 268 . . . 4 (((y = a, b xAa xBb) x C) ↔ (y = a, b (xAa x C) (xBb x C)))
982exbii 1583 . . 3 (ab((y = a, b xAa xBb) x C) ↔ ab(y = a, b (xAa x C) (xBb x C)))
10 brtxp 5783 . . . . 5 (x(AB)yab(y = a, b xAa xBb))
1110anbi1i 676 . . . 4 ((x(AB)y x C) ↔ (ab(y = a, b xAa xBb) x C))
12 brres 4949 . . . 4 (x((AB) C)y ↔ (x(AB)y x C))
13 19.41vv 1902 . . . 4 (ab((y = a, b xAa xBb) x C) ↔ (ab(y = a, b xAa xBb) x C))
1411, 12, 133bitr4i 268 . . 3 (x((AB) C)yab((y = a, b xAa xBb) x C))
15 brtxp 5783 . . . 4 (x((A C) ⊗ (B C))yab(y = a, b x(A C)a x(B C)b))
16 biid 227 . . . . . 6 (y = a, by = a, b)
17 brres 4949 . . . . . 6 (x(A C)a ↔ (xAa x C))
18 brres 4949 . . . . . 6 (x(B C)b ↔ (xBb x C))
1916, 17, 183anbi123i 1140 . . . . 5 ((y = a, b x(A C)a x(B C)b) ↔ (y = a, b (xAa x C) (xBb x C)))
20192exbii 1583 . . . 4 (ab(y = a, b x(A C)a x(B C)b) ↔ ab(y = a, b (xAa x C) (xBb x C)))
2115, 20bitri 240 . . 3 (x((A C) ⊗ (B C))yab(y = a, b (xAa x C) (xBb x C)))
229, 14, 213bitr4i 268 . 2 (x((AB) C)yx((A C) ⊗ (B C))y)
2322eqbrriv 4851 1 ((AB) C) = ((A C) ⊗ (B C))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561   class class class wbr 4639   ↾ cres 4774   ⊗ ctxp 5735 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-xp 4784  df-cnv 4785  df-res 4788  df-2nd 4797  df-txp 5736 This theorem is referenced by: (None)
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