Theorem xpcdaen 3726

Description: Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142.

Hypotheses

Ref	Expression
cdacomen.1	⊢ A ∈ V
cdacomen.2	⊢ B ∈ V
cdaassen.3	⊢ C ∈ V

Assertion

Ref	Expression
xpcdaen	⊢ (A × (B +c C)) ≈ ((A × B) +c (A × C))

Proof of Theorem xpcdaen

Step	Hyp	Ref	Expression
1		cdacomen.1	. . . 4 ⊢ A ∈ V
2		cdacomen.2	. . . . . 6 ⊢ B ∈ V
3		p0ex 1885	. . . . . 6 ⊢ {∅} ∈ V
4	2, 3	xpex 2488	. . . . 5 ⊢ (B × {∅}) ∈ V
5		cdaassen.3	. . . . . 6 ⊢ C ∈ V
6		snex 1859	. . . . . 6 ⊢ {1o} ∈ V
7	5, 6	xpex 2488	. . . . 5 ⊢ (C × {1o}) ∈ V
8	4, 7	unex 1949	. . . 4 ⊢ ((B × {∅}) ∪ (C × {1o})) ∈ V
9	1, 8	xpex 2488	. . 3 ⊢ (A × ((B × {∅}) ∪ (C × {1o}))) ∈ V
10	1, 2, 3	xpassen 3344	. . . . . 6 ⊢ ((A × B) × {∅}) ≈ (A × (B × {∅}))
11	1, 5, 6	xpassen 3344	. . . . . 6 ⊢ ((A × C) × {1o}) ≈ (A × (C × {1o}))
12	10, 11	pm3.2i 234	. . . . 5 ⊢ (((A × B) × {∅}) ≈ (A × (B × {∅})) ∧ ((A × C) × {1o}) ≈ (A × (C × {1o})))
13		0ne1oOLD 3113	. . . . . . 7 ⊢ ¬ ∅ = 1o
14		xpsndisj 2655	. . . . . . 7 ⊢ (¬ ∅ = 1o → (((A × B) × {∅}) ∩ ((A × C) × {1o})) = ∅)
15	13, 14	ax-mp 6	. . . . . 6 ⊢ (((A × B) × {∅}) ∩ ((A × C) × {1o})) = ∅
16		xpsndisj 2655	. . . . . . . . 9 ⊢ (¬ ∅ = 1o → ((B × {∅}) ∩ (C × {1o})) = ∅)
17	13, 16	ax-mp 6	. . . . . . . 8 ⊢ ((B × {∅}) ∩ (C × {1o})) = ∅
18		xpeq2 2441	. . . . . . . 8 ⊢ (((B × {∅}) ∩ (C × {1o})) = ∅ → (A × ((B × {∅}) ∩ (C × {1o}))) = (A × ∅))
19	17, 18	ax-mp 6	. . . . . . 7 ⊢ (A × ((B × {∅}) ∩ (C × {1o}))) = (A × ∅)
20		xpindi 2497	. . . . . . 7 ⊢ (A × ((B × {∅}) ∩ (C × {1o}))) = ((A × (B × {∅})) ∩ (A × (C × {1o})))
21		xp0 2652	. . . . . . 7 ⊢ (A × ∅) = ∅
22	19, 20, 21	3eqtr3 1124	. . . . . 6 ⊢ ((A × (B × {∅})) ∩ (A × (C × {1o}))) = ∅
23	15, 22	pm3.2i 234	. . . . 5 ⊢ ((((A × B) × {∅}) ∩ ((A × C) × {1o})) = ∅ ∧ ((A × (B × {∅})) ∩ (A × (C × {1o}))) = ∅)
24		unen 3338	. . . . 5 ⊢ (((((A × B) × {∅}) ≈ (A × (B × {∅})) ∧ ((A × C) × {1o}) ≈ (A × (C × {1o}))) ∧ ((((A × B) × {∅}) ∩ ((A × C) × {1o})) = ∅ ∧ ((A × (B × {∅})) ∩ (A × (C × {1o}))) = ∅)) → (((A × B) × {∅}) ∪ ((A × C) × {1o})) ≈ ((A × (B × {∅})) ∪ (A × (C × {1o}))))
25	12, 23, 24	mp2an 520	. . . 4 ⊢ (((A × B) × {∅}) ∪ ((A × C) × {1o})) ≈ ((A × (B × {∅})) ∪ (A × (C × {1o})))
26		xpundi 2461	. . . 4 ⊢ (A × ((B × {∅}) ∪ (C × {1o}))) = ((A × (B × {∅})) ∪ (A × (C × {1o})))
27	25, 26	breqtrr 2082	. . 3 ⊢ (((A × B) × {∅}) ∪ ((A × C) × {1o})) ≈ (A × ((B × {∅}) ∪ (C × {1o})))
28	9, 27	ensymi 3318	. 2 ⊢ (A × ((B × {∅}) ∪ (C × {1o}))) ≈ (((A × B) × {∅}) ∪ ((A × C) × {1o}))
29	2, 5	cdaval 3717	. . 3 ⊢ (B +c C) = ((B × {∅}) ∪ (C × {1o}))
30		xpeq2 2441	. . 3 ⊢ ((B +c C) = ((B × {∅}) ∪ (C × {1o})) → (A × (B +c C)) = (A × ((B × {∅}) ∪ (C × {1o}))))
31	29, 30	ax-mp 6	. 2 ⊢ (A × (B +c C)) = (A × ((B × {∅}) ∪ (C × {1o})))
32	1, 2	xpex 2488	. . 3 ⊢ (A × B) ∈ V
33	1, 5	xpex 2488	. . 3 ⊢ (A × C) ∈ V
34	32, 33	cdaval 3717	. 2 ⊢ ((A × B) +c (A × C)) = (((A × B) × {∅}) ∪ ((A × C) × {1o}))
35	28, 31, 34	3brtr4 2085	1 ⊢ (A × (B +c C)) ≈ ((A × B) +c (A × C))

Syntax hints:  ¬ wn 1   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  {csn 1808   class class class wbr 2054   × cxp 2408  (class class class)co 3001  1_oc1o 3099   ≈ cen 3271   +_c ccda 3714

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-suc 2205  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-opr 3003  df-oprab 3004  df-1o 3104  df-er 3200  df-en 3274  df-cda 3715

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