Theorem xpdom1g 4444

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 2623	. . . 4 |- (y = B -> (A ~<_ y <-> A ~<_ B))
2		xpeq1 3200	. . . . 5 |- (y = B -> (y X. z) = (B X. z))
3	2	breq2d 2630	. . . 4 |- (y = B -> ((A X. z) ~<_ (y X. z) <-> (A X. z) ~<_ (B X. z)))
4	1, 3	imbi12d 626	. . 3 |- (y = B -> ((A ~<_ y -> (A X. z) ~<_ (y X. z)) <-> (A ~<_ B -> (A X. z) ~<_ (B X. z))))
5		xpeq2 3201	. . . . 5 |- (z = C -> (A X. z) = (A X. C))
6		xpeq2 3201	. . . . 5 |- (z = C -> (B X. z) = (B X. C))
7	5, 6	breq12d 2631	. . . 4 |- (z = C -> ((A X. z) ~<_ (B X. z) <-> (A X. C) ~<_ (B X. C)))
8	7	imbi2d 612	. . 3 |- (z = C -> ((A ~<_ B -> (A X. z) ~<_ (B X. z)) <-> (A ~<_ B -> (A X. C) ~<_ (B X. C))))
9		visset 1813	. . . 4 |- y e. V
10		visset 1813	. . . 4 |- z e. V
11	9, 10	xpdom1 4443	. . 3 |- (A ~<_ y -> (A X. z) ~<_ (y X. z))
12	4, 8, 11	vtocl2g 1850	. 2 |- ((B e. R /\ C e. S) -> (A ~<_ B -> (A X. C) ~<_ (B X. C)))
13	12	3impia 830	1 |- ((B e. R /\ C e. S /\ A ~<_ B) -> (A X. C) ~<_ (B X. C))

Syntax hints:   -> wi 3   /\ w3a 775   = wceq 956   e. wcel 958   class class class wbr 2619   X. cxp 3168   ~<_ cdom 4365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-en 4368  df-dom 4369

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