Theorem xpen 3383

Description: Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254.

Hypotheses

Ref	Expression
xpen.1	⊢ A ∈ V
xpen.2	⊢ B ∈ V
xpen.3	⊢ C ∈ V
xpen.4	⊢ D ∈ V

Assertion

Ref	Expression
xpen	⊢ ((A ≈ B ∧ C ≈ D) → (A × C) ≈ (B × D))

Proof of Theorem xpen

Step	Hyp	Ref	Expression
1		entrt 3319	. 2 ⊢ (((A × C) ≈ (B × C) ∧ (B × C) ≈ (B × D)) → (A × C) ≈ (B × D))
2		xpen.1	. . . . . 6 ⊢ A ∈ V
3		xpen.2	. . . . . 6 ⊢ B ∈ V
4		xpen.3	. . . . . 6 ⊢ C ∈ V
5	2, 3, 4	xpdom2 3345	. . . . 5 ⊢ (A ≼ B → (C × A) ≼ (C × B))
6	3, 2, 4	xpdom2 3345	. . . . 5 ⊢ (B ≼ A → (C × B) ≼ (C × A))
7	5, 6	anim12i 268	. . . 4 ⊢ ((A ≼ B ∧ B ≼ A) → ((C × A) ≼ (C × B) ∧ (C × B) ≼ (C × A)))
8		sbthbg 3360	. . . . 5 ⊢ (B ∈ V → ((A ≼ B ∧ B ≼ A) ↔ A ≈ B))
9	3, 8	ax-mp 6	. . . 4 ⊢ ((A ≼ B ∧ B ≼ A) ↔ A ≈ B)
10	4, 3	xpex 2488	. . . . 5 ⊢ (C × B) ∈ V
11		sbthbg 3360	. . . . 5 ⊢ ((C × B) ∈ V → (((C × A) ≼ (C × B) ∧ (C × B) ≼ (C × A)) ↔ (C × A) ≈ (C × B)))
12	10, 11	ax-mp 6	. . . 4 ⊢ (((C × A) ≼ (C × B) ∧ (C × B) ≼ (C × A)) ↔ (C × A) ≈ (C × B))
13	7, 9, 12	3imtr3 191	. . 3 ⊢ (A ≈ B → (C × A) ≈ (C × B))
14	3, 4	xpex 2488	. . . . 5 ⊢ (B × C) ∈ V
15	4, 3	xpcomen 3343	. . . . 5 ⊢ (C × B) ≈ (B × C)
16		enen2 3376	. . . . 5 ⊢ (((B × C) ∈ V ∧ (C × B) ≈ (B × C)) → ((C × A) ≈ (C × B) ↔ (C × A) ≈ (B × C)))
17	14, 15, 16	mp2an 520	. . . 4 ⊢ ((C × A) ≈ (C × B) ↔ (C × A) ≈ (B × C))
18	2, 4	xpex 2488	. . . . 5 ⊢ (A × C) ∈ V
19	4, 2	xpcomen 3343	. . . . 5 ⊢ (C × A) ≈ (A × C)
20		enen1 3375	. . . . 5 ⊢ (((A × C) ∈ V ∧ (C × A) ≈ (A × C)) → ((C × A) ≈ (B × C) ↔ (A × C) ≈ (B × C)))
21	18, 19, 20	mp2an 520	. . . 4 ⊢ ((C × A) ≈ (B × C) ↔ (A × C) ≈ (B × C))
22	17, 21	bitr 151	. . 3 ⊢ ((C × A) ≈ (C × B) ↔ (A × C) ≈ (B × C))
23	13, 22	sylib 173	. 2 ⊢ (A ≈ B → (A × C) ≈ (B × C))
24		xpen.4	. . . . 5 ⊢ D ∈ V
25	4, 24, 3	xpdom2 3345	. . . 4 ⊢ (C ≼ D → (B × C) ≼ (B × D))
26	24, 4, 3	xpdom2 3345	. . . 4 ⊢ (D ≼ C → (B × D) ≼ (B × C))
27	25, 26	anim12i 268	. . 3 ⊢ ((C ≼ D ∧ D ≼ C) → ((B × C) ≼ (B × D) ∧ (B × D) ≼ (B × C)))
28		sbthbg 3360	. . . 4 ⊢ (D ∈ V → ((C ≼ D ∧ D ≼ C) ↔ C ≈ D))
29	24, 28	ax-mp 6	. . 3 ⊢ ((C ≼ D ∧ D ≼ C) ↔ C ≈ D)
30	3, 24	xpex 2488	. . . 4 ⊢ (B × D) ∈ V
31		sbthbg 3360	. . . 4 ⊢ ((B × D) ∈ V → (((B × C) ≼ (B × D) ∧ (B × D) ≼ (B × C)) ↔ (B × C) ≈ (B × D)))
32	30, 31	ax-mp 6	. . 3 ⊢ (((B × C) ≼ (B × D) ∧ (B × D) ≼ (B × C)) ↔ (B × C) ≈ (B × D))
33	27, 29, 32	3imtr3 191	. 2 ⊢ (C ≈ D → (B × C) ≈ (B × D))
34	1, 23, 33	syl2an 349	1 ⊢ ((A ≈ B ∧ C ≈ D) → (A × C) ≈ (B × D))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054   × cxp 2408   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  unxpdom2 3651  sucxpdom 3652  cdaassen 3725  xpomen 4928  qnnen 4931  infxpidmlem1 4933  infxpidmlem10 4942  infxpidmlem12 4944

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-er 3200  df-en 3274  df-dom 3275

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