Theorem tcdi 6165

Description: T raising distributes over addition. (Contributed by SF, 2-Mar-2015.)

Assertion

Ref	Expression
tcdi	|- ((A e. NC /\ B e. NC ) -> Tc (A +o B) = ( Tc A +o Tc B))

Proof of Theorem tcdi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eeanv 1913	. . 3 |- (E.xE.y(A = Nc x /\ B = Nc y) <-> (E.x A = Nc x /\ E.y B = Nc y))
2		vex 2863	. . . . . . . 8 |- x e. _V
3		0ex 4111	. . . . . . . . 9 |- (/) e. _V
4	3	complex 4105	. . . . . . . 8 |- ∼ (/) e. _V
5	2, 4	xpsnen 6050	. . . . . . 7 |- (x X. { ∼ (/)}) ~~ x
6		snex 4112	. . . . . . . . 9 |- { ∼ (/)} e. _V
7	2, 6	xpex 5116	. . . . . . . 8 |- (x X. { ∼ (/)}) e. _V
8	7	eqnc 6128	. . . . . . 7 |- ( Nc (x X. { ∼ (/)}) = Nc x <-> (x X. { ∼ (/)}) ~~ x)
9	5, 8	mpbir 200	. . . . . 6 |- Nc (x X. { ∼ (/)}) = Nc x
10	9	eqeq2i 2363	. . . . 5 |- (A = Nc (x X. { ∼ (/)}) <-> A = Nc x)
11		vex 2863	. . . . . . . 8 |- y e. _V
12	11, 3	xpsnen 6050	. . . . . . 7 |- (y X. {(/)}) ~~ y
13		snex 4112	. . . . . . . . 9 |- {(/)} e. _V
14	11, 13	xpex 5116	. . . . . . . 8 |- (y X. {(/)}) e. _V
15	14	eqnc 6128	. . . . . . 7 |- ( Nc (y X. {(/)}) = Nc y <-> (y X. {(/)}) ~~ y)
16	12, 15	mpbir 200	. . . . . 6 |- Nc (y X. {(/)}) = Nc y
17	16	eqeq2i 2363	. . . . 5 |- (B = Nc (y X. {(/)}) <-> B = Nc y)
18	10, 17	anbi12i 678	. . . 4 |- ((A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) <-> (A = Nc x /\ B = Nc y))
19	18	2exbii 1583	. . 3 |- (E.xE.y(A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) <-> E.xE.y(A = Nc x /\ B = Nc y))
20		elncs 6120	. . . 4 |- (A e. NC <-> E.x A = Nc x)
21		elncs 6120	. . . 4 |- (B e. NC <-> E.y B = Nc y)
22	20, 21	anbi12i 678	. . 3 |- ((A e. NC /\ B e. NC ) <-> (E.x A = Nc x /\ E.y B = Nc y))
23	1, 19, 22	3bitr4ri 269	. 2 |- ((A e. NC /\ B e. NC ) <-> E.xE.y(A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})))
24	7	ncelncsi 6122	. . . . . . 7 |- Nc (x X. { ∼ (/)}) e. NC
25	14	ncelncsi 6122	. . . . . . 7 |- Nc (y X. {(/)}) e. NC
26		ncaddccl 6145	. . . . . . 7 |- (( Nc (x X. { ∼ (/)}) e. NC /\ Nc (y X. {(/)}) e. NC ) -> ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) e. NC )
27	24, 25, 26	mp2an 653	. . . . . 6 |- ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) e. NC
28		tccl 6161	. . . . . 6 |- (( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) e. NC -> Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) e. NC )
29	27, 28	ax-mp 5	. . . . 5 |- Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) e. NC
30		tccl 6161	. . . . . . 7 |- ( Nc (x X. { ∼ (/)}) e. NC -> Tc Nc (x X. { ∼ (/)}) e. NC )
31	24, 30	ax-mp 5	. . . . . 6 |- Tc Nc (x X. { ∼ (/)}) e. NC
32		tccl 6161	. . . . . . 7 |- ( Nc (y X. {(/)}) e. NC -> Tc Nc (y X. {(/)}) e. NC )
33	25, 32	ax-mp 5	. . . . . 6 |- Tc Nc (y X. {(/)}) e. NC
34		ncaddccl 6145	. . . . . 6 |- (( Tc Nc (x X. { ∼ (/)}) e. NC /\ Tc Nc (y X. {(/)}) e. NC ) -> ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)})) e. NC )
35	31, 33, 34	mp2an 653	. . . . 5 |- ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)})) e. NC
36	7	ncid 6124	. . . . . . 7 |- (x X. { ∼ (/)}) e. Nc (x X. { ∼ (/)})
37	14	ncid 6124	. . . . . . 7 |- (y X. {(/)}) e. Nc (y X. {(/)})
38		necompl 3545	. . . . . . . 8 |- ∼ (/) =/= (/)
39	4, 38	xpnedisj 5514	. . . . . . 7 |- ((x X. { ∼ (/)}) i^i (y X. {(/)})) = (/)
40		eladdci 4400	. . . . . . 7 |- (((x X. { ∼ (/)}) e. Nc (x X. { ∼ (/)}) /\ (y X. {(/)}) e. Nc (y X. {(/)}) /\ ((x X. { ∼ (/)}) i^i (y X. {(/)})) = (/)) -> ((x X. { ∼ (/)}) u. (y X. {(/)})) e. ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})))
41	36, 37, 39, 40	mp3an 1277	. . . . . 6 |- ((x X. { ∼ (/)}) u. (y X. {(/)})) e. ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)}))
42		pw1eltc 6163	. . . . . 6 |- ((( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) e. NC /\ ((x X. { ∼ (/)}) u. (y X. {(/)})) e. ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)}))) -> ~P1 ((x X. { ∼ (/)}) u. (y X. {(/)})) e. Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})))
43	27, 41, 42	mp2an 653	. . . . 5 |- ~P1 ((x X. { ∼ (/)}) u. (y X. {(/)})) e. Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)}))
44		pw1un 4164	. . . . . 6 |- ~P1 ((x X. { ∼ (/)}) u. (y X. {(/)})) = (~P1 (x X. { ∼ (/)}) u. ~P1 (y X. {(/)}))
45		pw1eltc 6163	. . . . . . . 8 |- (( Nc (x X. { ∼ (/)}) e. NC /\ (x X. { ∼ (/)}) e. Nc (x X. { ∼ (/)})) -> ~P1 (x X. { ∼ (/)}) e. Tc Nc (x X. { ∼ (/)}))
46	24, 36, 45	mp2an 653	. . . . . . 7 |- ~P1 (x X. { ∼ (/)}) e. Tc Nc (x X. { ∼ (/)})
47		pw1eltc 6163	. . . . . . . 8 |- (( Nc (y X. {(/)}) e. NC /\ (y X. {(/)}) e. Nc (y X. {(/)})) -> ~P1 (y X. {(/)}) e. Tc Nc (y X. {(/)}))
48	25, 37, 47	mp2an 653	. . . . . . 7 |- ~P1 (y X. {(/)}) e. Tc Nc (y X. {(/)})
49		pw1eq 4144	. . . . . . . . 9 |- (((x X. { ∼ (/)}) i^i (y X. {(/)})) = (/) -> ~P1 ((x X. { ∼ (/)}) i^i (y X. {(/)})) = ~P1 (/))
50	39, 49	ax-mp 5	. . . . . . . 8 |- ~P1 ((x X. { ∼ (/)}) i^i (y X. {(/)})) = ~P1 (/)
51		pw1in 4165	. . . . . . . 8 |- ~P1 ((x X. { ∼ (/)}) i^i (y X. {(/)})) = (~P1 (x X. { ∼ (/)}) i^i ~P1 (y X. {(/)}))
52		pw10 4162	. . . . . . . 8 |- ~P1 (/) = (/)
53	50, 51, 52	3eqtr3i 2381	. . . . . . 7 |- (~P1 (x X. { ∼ (/)}) i^i ~P1 (y X. {(/)})) = (/)
54		eladdci 4400	. . . . . . 7 |- ((~P1 (x X. { ∼ (/)}) e. Tc Nc (x X. { ∼ (/)}) /\ ~P1 (y X. {(/)}) e. Tc Nc (y X. {(/)}) /\ (~P1 (x X. { ∼ (/)}) i^i ~P1 (y X. {(/)})) = (/)) -> (~P1 (x X. { ∼ (/)}) u. ~P1 (y X. {(/)})) e. ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)})))
55	46, 48, 53, 54	mp3an 1277	. . . . . 6 |- (~P1 (x X. { ∼ (/)}) u. ~P1 (y X. {(/)})) e. ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)}))
56	44, 55	eqeltri 2423	. . . . 5 |- ~P1 ((x X. { ∼ (/)}) u. (y X. {(/)})) e. ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)}))
57		nceleq 6150	. . . . 5 |- ((( Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) e. NC /\ ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)})) e. NC ) /\ (~P1 ((x X. { ∼ (/)}) u. (y X. {(/)})) e. Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) /\ ~P1 ((x X. { ∼ (/)}) u. (y X. {(/)})) e. ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)})))) -> Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) = ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)})))
58	29, 35, 43, 56, 57	mp4an 654	. . . 4 |- Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) = ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)}))
59		addceq12 4386	. . . . 5 |- ((A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) -> (A +o B) = ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})))
60		tceq 6159	. . . . 5 |- ((A +o B) = ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})) -> Tc (A +o B) = Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})))
61	59, 60	syl 15	. . . 4 |- ((A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) -> Tc (A +o B) = Tc ( Nc (x X. { ∼ (/)}) +o Nc (y X. {(/)})))
62		tceq 6159	. . . . . 6 |- (A = Nc (x X. { ∼ (/)}) -> Tc A = Tc Nc (x X. { ∼ (/)}))
63	62	adantr 451	. . . . 5 |- ((A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) -> Tc A = Tc Nc (x X. { ∼ (/)}))
64		tceq 6159	. . . . . 6 |- (B = Nc (y X. {(/)}) -> Tc B = Tc Nc (y X. {(/)}))
65	64	adantl 452	. . . . 5 |- ((A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) -> Tc B = Tc Nc (y X. {(/)}))
66	63, 65	addceq12d 4392	. . . 4 |- ((A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) -> ( Tc A +o Tc B) = ( Tc Nc (x X. { ∼ (/)}) +o Tc Nc (y X. {(/)})))
67	58, 61, 66	3eqtr4a 2411	. . 3 |- ((A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) -> Tc (A +o B) = ( Tc A +o Tc B))
68	67	exlimivv 1635	. 2 |- (E.xE.y(A = Nc (x X. { ∼ (/)}) /\ B = Nc (y X. {(/)})) -> Tc (A +o B) = ( Tc A +o Tc B))
69	23, 68	sylbi 187	1 |- ((A e. NC /\ B e. NC ) -> Tc (A +o B) = ( Tc A +o Tc B))

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   ∼ ccompl 3206   u. cun 3208   i^i cin 3209  (/)c0 3551  {csn 3738  ~P₁ cpw1 4136   +o cplc 4376   class class class wbr 4640   X. cxp 4771   ~~ cen 6029   NC cncs 6089   Nc cnc 6092   T_c ctc 6094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102  df-tc 6104

This theorem is referenced by:  tc2c  6167  tlecg  6231  nmembers1  6272  nchoicelem1  6290  nchoicelem2  6291  nchoicelem17  6306