Theorem xpeng 15691

Description: Equinumerosity law for cross product.

Assertion

Ref	Expression
xpeng	|- (((A e. W /\ B e. X) /\ (C e. Y /\ D e. Z)) -> ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D)))

Proof of Theorem xpeng

Step	Hyp	Ref	Expression
1		breq1 3341	. . . . . 6 |- (a = A -> (a ~~ b <-> A ~~ b))
2	1	anbi1d 679	. . . . 5 |- (a = A -> ((a ~~ b /\ C ~~ D) <-> (A ~~ b /\ C ~~ D)))
3		xpeq1 4016	. . . . . 6 |- (a = A -> (a X. C) = (A X. C))
4	3	breq1d 3348	. . . . 5 |- (a = A -> ((a X. C) ~~ (b X. D) <-> (A X. C) ~~ (b X. D)))
5	2, 4	imbi12d 688	. . . 4 |- (a = A -> (((a ~~ b /\ C ~~ D) -> (a X. C) ~~ (b X. D)) <-> ((A ~~ b /\ C ~~ D) -> (A X. C) ~~ (b X. D))))
6	5	imbi2d 674	. . 3 |- (a = A -> (((C e. Y /\ D e. Z) -> ((a ~~ b /\ C ~~ D) -> (a X. C) ~~ (b X. D))) <-> ((C e. Y /\ D e. Z) -> ((A ~~ b /\ C ~~ D) -> (A X. C) ~~ (b X. D)))))
7		breq2 3342	. . . . . 6 |- (b = B -> (A ~~ b <-> A ~~ B))
8	7	anbi1d 679	. . . . 5 |- (b = B -> ((A ~~ b /\ C ~~ D) <-> (A ~~ B /\ C ~~ D)))
9		xpeq1 4016	. . . . . 6 |- (b = B -> (b X. D) = (B X. D))
10	9	breq2d 3350	. . . . 5 |- (b = B -> ((A X. C) ~~ (b X. D) <-> (A X. C) ~~ (B X. D)))
11	8, 10	imbi12d 688	. . . 4 |- (b = B -> (((A ~~ b /\ C ~~ D) -> (A X. C) ~~ (b X. D)) <-> ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))))
12	11	imbi2d 674	. . 3 |- (b = B -> (((C e. Y /\ D e. Z) -> ((A ~~ b /\ C ~~ D) -> (A X. C) ~~ (b X. D))) <-> ((C e. Y /\ D e. Z) -> ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D)))))
13		breq1 3341	. . . . . 6 |- (c = C -> (c ~~ d <-> C ~~ d))
14	13	anbi2d 678	. . . . 5 |- (c = C -> ((a ~~ b /\ c ~~ d) <-> (a ~~ b /\ C ~~ d)))
15		xpeq2 4017	. . . . . 6 |- (c = C -> (a X. c) = (a X. C))
16	15	breq1d 3348	. . . . 5 |- (c = C -> ((a X. c) ~~ (b X. d) <-> (a X. C) ~~ (b X. d)))
17	14, 16	imbi12d 688	. . . 4 |- (c = C -> (((a ~~ b /\ c ~~ d) -> (a X. c) ~~ (b X. d)) <-> ((a ~~ b /\ C ~~ d) -> (a X. C) ~~ (b X. d))))
18		breq2 3342	. . . . . 6 |- (d = D -> (C ~~ d <-> C ~~ D))
19	18	anbi2d 678	. . . . 5 |- (d = D -> ((a ~~ b /\ C ~~ d) <-> (a ~~ b /\ C ~~ D)))
20		xpeq2 4017	. . . . . 6 |- (d = D -> (b X. d) = (b X. D))
21	20	breq2d 3350	. . . . 5 |- (d = D -> ((a X. C) ~~ (b X. d) <-> (a X. C) ~~ (b X. D)))
22	19, 21	imbi12d 688	. . . 4 |- (d = D -> (((a ~~ b /\ C ~~ d) -> (a X. C) ~~ (b X. d)) <-> ((a ~~ b /\ C ~~ D) -> (a X. C) ~~ (b X. D))))
23		visset 2295	. . . . 5 |- a e. _V
24		visset 2295	. . . . 5 |- b e. _V
25		visset 2295	. . . . 5 |- c e. _V
26		visset 2295	. . . . 5 |- d e. _V
27	23, 24, 25, 26	xpen 5582	. . . 4 |- ((a ~~ b /\ c ~~ d) -> (a X. c) ~~ (b X. d))
28	17, 22, 27	vtocl2g 2349	. . 3 |- ((C e. Y /\ D e. Z) -> ((a ~~ b /\ C ~~ D) -> (a X. C) ~~ (b X. D)))
29	6, 12, 28	vtocl2g 2349	. 2 |- ((A e. W /\ B e. X) -> ((C e. Y /\ D e. Z) -> ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))))
30	29	imp 377	1 |- (((A e. W /\ B e. X) /\ (C e. Y /\ D e. Z)) -> ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338   X. cxp 3984   ~~ cen 5423

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428

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