Theorem xpdom1 4449

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

Hypotheses

Ref	Expression
xpdom.1	⊢ B ∈ V
xpdom.2	⊢ C ∈ V

Assertion

Ref	Expression
xpdom1	⊢ (A ≼ B → (A × C) ≼ (B × C))

Proof of Theorem xpdom1

Step	Hyp	Ref	Expression
1		endomtr 4426	. . 3 ⊢ (((A × C) ≈ (C × A) ⋀ (C × A) ≼ (C × B)) → (A × C) ≼ (C × B))
2		reldom 4379	. . . . 5 ⊢ Rel ≼
3	2	brrelexi 3214	. . . 4 ⊢ (A ≼ B → A ∈ V)
4		xpdom.2	. . . . 5 ⊢ C ∈ V
5		xpcomeng 4446	. . . . 5 ⊢ ((A ∈ V ⋀ C ∈ V) → (A × C) ≈ (C × A))
6	4, 5	mpan2 698	. . . 4 ⊢ (A ∈ V → (A × C) ≈ (C × A))
7	3, 6	syl 10	. . 3 ⊢ (A ≼ B → (A × C) ≈ (C × A))
8		xpdom.1	. . . 4 ⊢ B ∈ V
9	8, 4	xpdom2 4448	. . 3 ⊢ (A ≼ B → (C × A) ≼ (C × B))
10	1, 7, 9	sylanc 473	. 2 ⊢ (A ≼ B → (A × C) ≼ (C × B))
11	4, 8	xpcomen 4445	. . 3 ⊢ (C × B) ≈ (B × C)
12		domentr 4427	. . 3 ⊢ (((A × C) ≼ (C × B) ⋀ (C × B) ≈ (B × C)) → (A × C) ≼ (B × C))
13	11, 12	mpan2 698	. 2 ⊢ ((A × C) ≼ (C × B) → (A × C) ≼ (B × C))
14	10, 13	syl 10	1 ⊢ (A ≼ B → (A × C) ≼ (B × C))

Syntax hints:   → wi 3   ∈ wcel 960  Vcvv 1814   class class class wbr 2624   × cxp 3174   ≈ cen 4370   ≼ cdom 4371

This theorem is referenced by:  xpdom1g 4450  uniimadom 4820

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-en 4374  df-dom 4375

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