Theorem xpcdaen 5083

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 2828	. . . . . 6 |- {(/)} e. V
4	2, 3	xpex 3349	. . . . 5 |- (B X. {(/)}) e. V
5		cdaassen.3	. . . . . 6 |- C e. V
6		snex 2826	. . . . . 6 |- {1o} e. V
7	5, 6	xpex 3349	. . . . 5 |- (C X. {1o}) e. V
8	4, 7	unex 3095	. . . 4 |- ((B X. {(/)}) u. (C X. {1o})) e. V
9	1, 8	xpex 3349	. . 3 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) e. V
10	1, 2, 3	xpassen 4582	. . . . . 6 |- ((A X. B) X. {(/)}) ~~ (A X. (B X. {(/)}))
11	1, 5, 6	xpassen 4582	. . . . . 6 |- ((A X. C) X. {1o}) ~~ (A X. (C X. {1o}))
12	10, 11	pm3.2i 283	. . . . 5 |- (((A X. B) X. {(/)}) ~~ (A X. (B X. {(/)})) /\ ((A X. C) X. {1o}) ~~ (A X. (C X. {1o})))
13		xp01disj 4279	. . . . . 6 |- (((A X. B) X. {(/)}) i^i ((A X. C) X. {1o})) = (/)
14		xp01disj 4279	. . . . . . . 8 |- ((B X. {(/)}) i^i (C X. {1o})) = (/)
15	14	xpeq2i 3286	. . . . . . 7 |- (A X. ((B X. {(/)}) i^i (C X. {1o}))) = (A X. (/))
16		xpindi 3363	. . . . . . 7 |- (A X. ((B X. {(/)}) i^i (C X. {1o}))) = ((A X. (B X. {(/)})) i^i (A X. (C X. {1o})))
17		xp0 3550	. . . . . . 7 |- (A X. (/)) = (/)
18	15, 16, 17	3eqtr3i 1546	. . . . . 6 |- ((A X. (B X. {(/)})) i^i (A X. (C X. {1o}))) = (/)
19	13, 18	pm3.2i 283	. . . . 5 |- ((((A X. B) X. {(/)}) i^i ((A X. C) X. {1o})) = (/) /\ ((A X. (B X. {(/)})) i^i (A X. (C X. {1o}))) = (/))
20		unen 4575	. . . . 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}))))
21	12, 19, 20	mp2an 701	. . . 4 |- (((A X. B) X. {(/)}) u. ((A X. C) X. {1o})) ~~ ((A X. (B X. {(/)})) u. (A X. (C X. {1o})))
22		xpundi 3310	. . . 4 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) = ((A X. (B X. {(/)})) u. (A X. (C X. {1o})))
23	21, 22	breqtrri 2713	. . 3 |- (((A X. B) X. {(/)}) u. ((A X. C) X. {1o})) ~~ (A X. ((B X. {(/)}) u. (C X. {1o})))
24	9, 23	ensymi 4554	. 2 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) ~~ (((A X. B) X. {(/)}) u. ((A X. C) X. {1o}))
25	2, 5	cdavali 5070	. . 3 |- (B +c C) = ((B X. {(/)}) u. (C X. {1o}))
26	25	xpeq2i 3286	. 2 |- (A X. (B +c C)) = (A X. ((B X. {(/)}) u. (C X. {1o})))
27	1, 2	xpex 3349	. . 3 |- (A X. B) e. V
28	1, 5	xpex 3349	. . 3 |- (A X. C) e. V
29	27, 28	cdavali 5070	. 2 |- ((A X. B) +c (A X. C)) = (((A X. B) X. {(/)}) u. ((A X. C) X. {1o}))
30	24, 26, 29	3brtr4i 2716	1 |- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))

Syntax hints:   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857   u. cun 2097   i^i cin 2098  (/)c0 2332  {csn 2467   class class class wbr 2692   X. cxp 3249  (class class class)co 4021  1oc1o 4264   ~~ cen 4505   +c ccda 5067

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1o 4269  df-er 4401  df-en 4509  df-cda 5068

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