Theorem inxp 2496

Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
inxp	|- ((A X. B) i^i (C X. D)) = ((A i^i C) X. (B i^i D))

Proof of Theorem inxp

Step	Hyp	Ref	Expression
1		relxp 2486	. . 3 |- Rel (A X. B)
2		relin 2491	. . 3 |- (Rel (A X. B) -> Rel ((A X. B) i^i (C X. D)))
3	1, 2	ax-mp 6	. 2 |- Rel ((A X. B) i^i (C X. D))
4		relxp 2486	. 2 |- Rel ((A i^i C) X. (B i^i D))
5		an4 388	. . . 4 |- (((x e. A /\ y e. B) /\ (x e. C /\ y e. D)) <-> ((x e. A /\ x e. C) /\ (y e. B /\ y e. D)))
6		visset 1350	. . . . . 6 |- y e. V
7	6	opelxp 2452	. . . . 5 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
8	6	opelxp 2452	. . . . 5 |- (<.x, y>. e. (C X. D) <-> (x e. C /\ y e. D))
9	7, 8	anbi12i 369	. . . 4 |- ((<.x, y>. e. (A X. B) /\ <.x, y>. e. (C X. D)) <-> ((x e. A /\ y e. B) /\ (x e. C /\ y e. D)))
10		elin 1635	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
11		elin 1635	. . . . 5 |- (y e. (B i^i D) <-> (y e. B /\ y e. D))
12	10, 11	anbi12i 369	. . . 4 |- ((x e. (A i^i C) /\ y e. (B i^i D)) <-> ((x e. A /\ x e. C) /\ (y e. B /\ y e. D)))
13	5, 9, 12	3bitr4 158	. . 3 |- ((<.x, y>. e. (A X. B) /\ <.x, y>. e. (C X. D)) <-> (x e. (A i^i C) /\ y e. (B i^i D)))
14		elin 1635	. . 3 |- (<.x, y>. e. ((A X. B) i^i (C X. D)) <-> (<.x, y>. e. (A X. B) /\ <.x, y>. e. (C X. D)))
15	6	opelxp 2452	. . 3 |- (<.x, y>. e. ((A i^i C) X. (B i^i D)) <-> (x e. (A i^i C) /\ y e. (B i^i D)))
16	13, 14, 15	3bitr4 158	. 2 |- (<.x, y>. e. ((A X. B) i^i (C X. D)) <-> <.x, y>. e. ((A i^i C) X. (B i^i D)))
17	3, 4, 16	cleqreli 2484	1 |- ((A X. B) i^i (C X. D)) = ((A i^i C) X. (B i^i D))

Syntax hints:   /\ wa 196   = wceq 1091   e. wcel 1092   i^i cin 1486  <.cop 1810   X. cxp 2408  Rel wrel 2415

This theorem is referenced by:  xpindi 2497  xpindir 2498  resabs1 2592  xpdisj1 2653  xpdisj2 2654

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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