Theorem inxp 3275

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 3261	. . 3 |- Rel (A X. B)
2		relin1 3268	. . 3 |- (Rel (A X. B) -> Rel ((A X. B) i^i (C X. D)))
3	1, 2	ax-mp 7	. 2 |- Rel ((A X. B) i^i (C X. D))
4		relxp 3261	. 2 |- Rel ((A i^i C) X. (B i^i D))
5		an4 508	. . . 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 1816	. . . . . 6 |- y e. V
7	6	opelxp 3220	. . . . 5 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
8	6	opelxp 3220	. . . . 5 |- (<.x, y>. e. (C X. D) <-> (x e. C /\ y e. D))
9	7, 8	anbi12i 484	. . . 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 2210	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
11		elin 2210	. . . . 5 |- (y e. (B i^i D) <-> (y e. B /\ y e. D))
12	10, 11	anbi12i 484	. . . 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 183	. . 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 2210	. . 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 3220	. . 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 183	. 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	eqrelriv 3257	1 |- ((A X. B) i^i (C X. D)) = ((A i^i C) X. (B i^i D))

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960   i^i cin 2049  <.cop 2415   X. cxp 3174  Rel wrel 3181

This theorem is referenced by:  xpindi 3276  xpindir 3277  dmxpin 3340  xpdisj1 3474  xpdisj2 3475  rescnvcnv 3499  curry1 4104  metres 7820

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-rel 3191

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