Theorem tcdi 6165

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

Assertion

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

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∼ ccompl 3206   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  ℘₁cpw1 4136   +_c cplc 4376   class class class wbr 4640   × 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