Theorem xpcdaen 3726

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

Hypotheses

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

Assertion

Ref	Expression
xpcdaen	|- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))

Proof of Theorem xpcdaen

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

Syntax hints:  -. wn 1   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485   i^i cin 1486  (/)c0 1707  {csn 1808   class class class wbr 2054   X. cxp 2408  (class class class)co 3001  1oc1o 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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