Theorem xpcdaen 4931

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 2770	. . . . . 6 |- {(/)} e. V
4	2, 3	xpex 3260	. . . . 5 |- (B X. {(/)}) e. V
5		cdaassen.3	. . . . . 6 |- C e. V
6		snex 2750	. . . . . 6 |- {1o} e. V
7	5, 6	xpex 3260	. . . . 5 |- (C X. {1o}) e. V
8	4, 7	unex 2872	. . . 4 |- ((B X. {(/)}) u. (C X. {1o})) e. V
9	1, 8	xpex 3260	. . 3 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) e. V
10	1, 2, 3	xpassen 4441	. . . . . 6 |- ((A X. B) X. {(/)}) ~~ (A X. (B X. {(/)}))
11	1, 5, 6	xpassen 4441	. . . . . 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 4143	. . . . . 6 |- (((A X. B) X. {(/)}) i^i ((A X. C) X. {1o})) = (/)
14		xp01disj 4143	. . . . . . . 8 |- ((B X. {(/)}) i^i (C X. {1o})) = (/)
15		xpeq2 3201	. . . . . . . 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 3270	. . . . . . 7 |- (A X. ((B X. {(/)}) i^i (C X. {1o}))) = ((A X. (B X. {(/)})) i^i (A X. (C X. {1o})))
18		xp0 3465	. . . . . . 7 |- (A X. (/)) = (/)
19	16, 17, 18	3eqtr3 1503	. . . . . 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 4434	. . . . 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 697	. . . 4 |- (((A X. B) X. {(/)}) u. ((A X. C) X. {1o})) ~~ ((A X. (B X. {(/)})) u. (A X. (C X. {1o})))
23		xpundi 3225	. . . 4 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) = ((A X. (B X. {(/)})) u. (A X. (C X. {1o})))
24	22, 23	breqtrr 2640	. . 3 |- (((A X. B) X. {(/)}) u. ((A X. C) X. {1o})) ~~ (A X. ((B X. {(/)}) u. (C X. {1o})))
25	9, 24	ensymi 4413	. 2 |- (A X. ((B X. {(/)}) u. (C X. {1o}))) ~~ (((A X. B) X. {(/)}) u. ((A X. C) X. {1o}))
26	2, 5	cdaval 4920	. . 3 |- (B +c C) = ((B X. {(/)}) u. (C X. {1o}))
27		xpeq2 3201	. . 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 3260	. . 3 |- (A X. B) e. V
30	1, 5	xpex 3260	. . 3 |- (A X. C) e. V
31	29, 30	cdaval 4920	. 2 |- ((A X. B) +c (A X. C)) = (((A X. B) X. {(/)}) u. ((A X. C) X. {1o}))
32	25, 28, 31	3brtr4 2643	1 |- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045   i^i cin 2046  (/)c0 2280  {csn 2409   class class class wbr 2619   X. cxp 3168  (class class class)co 3963  1oc1o 4128   ~~ cen 4364   +c ccda 4917

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-1o 4133  df-er 4261  df-en 4368  df-cda 4918

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