Theorem xpdom1 3346

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

Hypotheses

Ref	Expression
xpdom2.1	|- A e. V
xpdom2.2	|- B e. V
xpdom2.3	|- C e. V

Assertion

Ref	Expression
xpdom1	|- (A ~<_ B -> (A X. C) ~<_ (B X. C))

Proof of Theorem xpdom1

Step	Hyp	Ref	Expression
1		xpdom2.1	. . 3 |- A e. V
2		xpdom2.2	. . 3 |- B e. V
3		xpdom2.3	. . 3 |- C e. V
4	1, 2, 3	xpdom2 3345	. 2 |- (A ~<_ B -> (C X. A) ~<_ (C X. B))
5	1, 3	xpcomen 3343	. . 3 |- (A X. C) ~~ (C X. A)
6		endomtr 3325	. . 3 |- (((A X. C) ~~ (C X. A) /\ (C X. A) ~<_ (C X. B)) -> (A X. C) ~<_ (C X. B))
7	5, 6	mpan 518	. 2 |- ((C X. A) ~<_ (C X. B) -> (A X. C) ~<_ (C X. B))
8	3, 2	xpcomen 3343	. . 3 |- (C X. B) ~~ (B X. C)
9		domentr 3326	. . 3 |- (((A X. C) ~<_ (C X. B) /\ (C X. B) ~~ (B X. C)) -> (A X. C) ~<_ (B X. C))
10	8, 9	mpan2 519	. 2 |- ((A X. C) ~<_ (C X. B) -> (A X. C) ~<_ (B X. C))
11	4, 7, 10	3syl 21	1 |- (A ~<_ B -> (A X. C) ~<_ (B X. C))

Syntax hints:   -> wi 2   e. wcel 1092  Vcvv 1348   class class class wbr 2054   X. cxp 2408   ~~ cen 3271   ~<_ cdom 3272

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-dom 3275

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