Theorem xpcdaen 4903

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 2760	. . . . . 6 |- {(/)} e. V
4	2, 3	xpex 3250	. . . . 5 |- (B X. {(/)}) e. V
5		cdaassen.3	. . . . . 6 |- C e. V
6		snex 2740	. . . . . 6 |- {1o} e. V
7	5, 6	xpex 3250	. . . . 5 |- (C X. {1o}) e. V
8	4, 7	unex 2863	. . . 4 |- ((B X. {(/)}) u. (C X. {1o})) e. V
9	1, 8	xpex 3250	. . 3 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) e. V
10	1, 2, 3	xpassen 4421	. . . . . 6 |- ((A X. B) X. {(/)}) ~~ (A X. (B X. {(/)}))
11	1, 5, 6	xpassen 4421	. . . . . 6 |- ((A X. C) X. {1o}) ~~ (A X. (C X. {1o}))
12	10, 11	pm3.2i 285	. . . . 5 |- (((A X. B) X. {(/)}) ~~ (A X. (B X. {(/)})) /\ ((A X. C) X. {1o}) ~~ (A X. (C X. {1o})))
13		xp01disj 4127	. . . . . 6 |- (((A X. B) X. {(/)}) i^i ((A X. C) X. {1o})) = (/)
14		xp01disj 4127	. . . . . . . 8 |- ((B X. {(/)}) i^i (C X. {1o})) = (/)
15		xpeq2 3191	. . . . . . . 8 |- (((B X. {(/)}) i^i (C X. {1o})) = (/) -> (A X. ((B X. {(/)}) i^i (C X. {1o}))) = (A X. (/)))
16	14, 15	ax-mp 7	. . . . . . 7 |- (A X. ((B X. {(/)}) i^i (C X. {1o}))) = (A X. (/))
17		xpindi 3260	. . . . . . 7 |- (A X. ((B X. {(/)}) i^i (C X. {1o}))) = ((A X. (B X. {(/)})) i^i (A X. (C X. {1o})))
18		xp0 3451	. . . . . . 7 |- (A X. (/)) = (/)
19	16, 17, 18	3eqtr3 1495	. . . . . 6 |- ((A X. (B X. {(/)})) i^i (A X. (C X. {1o}))) = (/)
20	13, 19	pm3.2i 285	. . . . 5 |- ((((A X. B) X. {(/)}) i^i ((A X. C) X. {1o})) = (/) /\ ((A X. (B X. {(/)})) i^i (A X. (C X. {1o}))) = (/))
21		unen 4414	. . . . 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}))))
22	12, 20, 21	mp2an 695	. . . 4 |- (((A X. B) X. {(/)}) u. ((A X. C) X. {1o})) ~~ ((A X. (B X. {(/)})) u. (A X. (C X. {1o})))
23		xpundi 3215	. . . 4 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) = ((A X. (B X. {(/)})) u. (A X. (C X. {1o})))
24	22, 23	breqtrr 2630	. . 3 |- (((A X. B) X. {(/)}) u. ((A X. C) X. {1o})) ~~ (A X. ((B X. {(/)}) u. (C X. {1o})))
25	9, 24	ensymi 4394	. 2 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) ~~ (((A X. B) X. {(/)}) u. ((A X. C) X. {1o}))
26	2, 5	cdaval 4892	. . 3 |- (B +c C) = ((B X. {(/)}) u. (C X. {1o}))
27		xpeq2 3191	. . 3 |- ((B +c C) = ((B X. {(/)}) u. (C X. {1o})) -> (A X. (B +c C)) = (A X. ((B X. {(/)}) u. (C X. {1o}))))
28	26, 27	ax-mp 7	. 2 |- (A X. (B +c C)) = (A X. ((B X. {(/)}) u. (C X. {1o})))
29	1, 2	xpex 3250	. . 3 |- (A X. B) e. V
30	1, 5	xpex 3250	. . 3 |- (A X. C) e. V
31	29, 30	cdaval 4892	. 2 |- ((A X. B) +c (A X. C)) = (((A X. B) X. {(/)}) u. ((A X. C) X. {1o}))
32	25, 28, 31	3brtr4 2633	1 |- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609   X. cxp 3158  (class class class)co 3948  1oc1o 4112   ~~ cen 4348   +c ccda 4889

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-oprab 3951  df-1o 4117  df-er 4245  df-en 4351  df-cda 4890

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