Theorem inxp 3264

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 3250	. . 3 |- Rel (A X. B)
2		relin1 3257	. . 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 3250	. 2 |- Rel ((A i^i C) X. (B i^i D))
5		an4 506	. . . 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 1809	. . . . . 6 |- y e. V
7	6	opelxp 3209	. . . . 5 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
8	6	opelxp 3209	. . . . 5 |- (<.x, y>. e. (C X. D) <-> (x e. C /\ y e. D))
9	7, 8	anbi12i 482	. . . 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 2203	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
11		elin 2203	. . . . 5 |- (y e. (B i^i D) <-> (y e. B /\ y e. D))
12	10, 11	anbi12i 482	. . . 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 2203	. . 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 3209	. . 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 3246	1 |- ((A X. B) i^i (C X. D)) = ((A i^i C) X. (B i^i D))

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956   i^i cin 2042  <.cop 2407   X. cxp 3163  Rel wrel 3170

This theorem is referenced by:  xpindi 3265  xpindir 3266  dmxpin 3329  xpdisj1 3460  xpdisj2 3461  rescnvcnv 3485  curry1 4088  metres 7775

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662  df-xp 3179  df-rel 3180

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