Theorem xpdom1g 4424

Description: Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149.

Assertion

Ref	Expression
xpdom1g	|- ((B e. R /\ C e. S /\ A ~<_ B) -> (A X. C) ~<_ (B X. C))

Proof of Theorem xpdom1g

Step	Hyp	Ref	Expression
1		breq2 2613	. . . 4 |- (y = B -> (A ~<_ y <-> A ~<_ B))
2		xpeq1 3190	. . . . 5 |- (y = B -> (y X. z) = (B X. z))
3	2	breq2d 2620	. . . 4 |- (y = B -> ((A X. z) ~<_ (y X. z) <-> (A X. z) ~<_ (B X. z)))
4	1, 3	imbi12d 624	. . 3 |- (y = B -> ((A ~<_ y -> (A X. z) ~<_ (y X. z)) <-> (A ~<_ B -> (A X. z) ~<_ (B X. z))))
5		xpeq2 3191	. . . . 5 |- (z = C -> (A X. z) = (A X. C))
6		xpeq2 3191	. . . . 5 |- (z = C -> (B X. z) = (B X. C))
7	5, 6	breq12d 2621	. . . 4 |- (z = C -> ((A X. z) ~<_ (B X. z) <-> (A X. C) ~<_ (B X. C)))
8	7	imbi2d 610	. . 3 |- (z = C -> ((A ~<_ B -> (A X. z) ~<_ (B X. z)) <-> (A ~<_ B -> (A X. C) ~<_ (B X. C))))
9		visset 1804	. . . 4 |- y e. V
10		visset 1804	. . . 4 |- z e. V
11	9, 10	xpdom1 4423	. . 3 |- (A ~<_ y -> (A X. z) ~<_ (y X. z))
12	4, 8, 11	vtocl2g 1841	. 2 |- ((B e. R /\ C e. S) -> (A ~<_ B -> (A X. C) ~<_ (B X. C)))
13	12	3impia 828	1 |- ((B e. R /\ C e. S /\ A ~<_ B) -> (A X. C) ~<_ (B X. C))

Syntax hints:   -> wi 3   /\ w3a 773   = wceq 953   e. wcel 955   class class class wbr 2609   X. cxp 3158   ~<_ cdom 4349

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-en 4351  df-dom 4352

Copyright terms: Public domain