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Theorem tcdi 6164
 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.
StepHypRef Expression
1 eeanv 1913 . . 3 (xy(A = Nc x B = Nc y) ↔ (x A = Nc x y B = Nc y))
2 vex 2862 . . . . . . . 8 x V
3 0ex 4110 . . . . . . . . 9 V
43complex 4104 . . . . . . . 8 V
52, 4xpsnen 6049 . . . . . . 7 (x × { ∼ }) ≈ x
6 snex 4111 . . . . . . . . 9 { ∼ } V
72, 6xpex 5115 . . . . . . . 8 (x × { ∼ }) V
87eqnc 6127 . . . . . . 7 ( Nc (x × { ∼ }) = Nc x ↔ (x × { ∼ }) ≈ x)
95, 8mpbir 200 . . . . . 6 Nc (x × { ∼ }) = Nc x
109eqeq2i 2363 . . . . 5 (A = Nc (x × { ∼ }) ↔ A = Nc x)
11 vex 2862 . . . . . . . 8 y V
1211, 3xpsnen 6049 . . . . . . 7 (y × {}) ≈ y
13 snex 4111 . . . . . . . . 9 {} V
1411, 13xpex 5115 . . . . . . . 8 (y × {}) V
1514eqnc 6127 . . . . . . 7 ( Nc (y × {}) = Nc y ↔ (y × {}) ≈ y)
1612, 15mpbir 200 . . . . . 6 Nc (y × {}) = Nc y
1716eqeq2i 2363 . . . . 5 (B = Nc (y × {}) ↔ B = Nc y)
1810, 17anbi12i 678 . . . 4 ((A = Nc (x × { ∼ }) B = Nc (y × {})) ↔ (A = Nc x B = Nc y))
19182exbii 1583 . . 3 (xy(A = Nc (x × { ∼ }) B = Nc (y × {})) ↔ xy(A = Nc x B = Nc y))
20 elncs 6119 . . . 4 (A NCx A = Nc x)
21 elncs 6119 . . . 4 (B NCy B = Nc y)
2220, 21anbi12i 678 . . 3 ((A NC B NC ) ↔ (x A = Nc x y B = Nc y))
231, 19, 223bitr4ri 269 . 2 ((A NC B NC ) ↔ xy(A = Nc (x × { ∼ }) B = Nc (y × {})))
247ncelncsi 6121 . . . . . . 7 Nc (x × { ∼ }) NC
2514ncelncsi 6121 . . . . . . 7 Nc (y × {}) NC
26 ncaddccl 6144 . . . . . . 7 (( Nc (x × { ∼ }) NC Nc (y × {}) NC ) → ( Nc (x × { ∼ }) +c Nc (y × {})) NC )
2724, 25, 26mp2an 653 . . . . . 6 ( Nc (x × { ∼ }) +c Nc (y × {})) NC
28 tccl 6160 . . . . . 6 (( Nc (x × { ∼ }) +c Nc (y × {})) NCTc ( Nc (x × { ∼ }) +c Nc (y × {})) NC )
2927, 28ax-mp 8 . . . . 5 Tc ( Nc (x × { ∼ }) +c Nc (y × {})) NC
30 tccl 6160 . . . . . . 7 ( Nc (x × { ∼ }) NCTc Nc (x × { ∼ }) NC )
3124, 30ax-mp 8 . . . . . 6 Tc Nc (x × { ∼ }) NC
32 tccl 6160 . . . . . . 7 ( Nc (y × {}) NCTc Nc (y × {}) NC )
3325, 32ax-mp 8 . . . . . 6 Tc Nc (y × {}) NC
34 ncaddccl 6144 . . . . . 6 (( Tc Nc (x × { ∼ }) NC Tc Nc (y × {}) NC ) → ( Tc Nc (x × { ∼ }) +c Tc Nc (y × {})) NC )
3531, 33, 34mp2an 653 . . . . 5 ( Tc Nc (x × { ∼ }) +c Tc Nc (y × {})) NC
367ncid 6123 . . . . . . 7 (x × { ∼ }) Nc (x × { ∼ })
3714ncid 6123 . . . . . . 7 (y × {}) Nc (y × {})
38 necompl 3544 . . . . . . . 8
394, 38xpnedisj 5513 . . . . . . 7 ((x × { ∼ }) ∩ (y × {})) =
40 eladdci 4399 . . . . . . 7 (((x × { ∼ }) Nc (x × { ∼ }) (y × {}) Nc (y × {}) ((x × { ∼ }) ∩ (y × {})) = ) → ((x × { ∼ }) ∪ (y × {})) ( Nc (x × { ∼ }) +c Nc (y × {})))
4136, 37, 39, 40mp3an 1277 . . . . . 6 ((x × { ∼ }) ∪ (y × {})) ( Nc (x × { ∼ }) +c Nc (y × {}))
42 pw1eltc 6162 . . . . . 6 ((( Nc (x × { ∼ }) +c Nc (y × {})) NC ((x × { ∼ }) ∪ (y × {})) ( Nc (x × { ∼ }) +c Nc (y × {}))) → 1((x × { ∼ }) ∪ (y × {})) Tc ( Nc (x × { ∼ }) +c Nc (y × {})))
4327, 41, 42mp2an 653 . . . . 5 1((x × { ∼ }) ∪ (y × {})) Tc ( Nc (x × { ∼ }) +c Nc (y × {}))
44 pw1un 4163 . . . . . 6 1((x × { ∼ }) ∪ (y × {})) = (1(x × { ∼ }) ∪ 1(y × {}))
45 pw1eltc 6162 . . . . . . . 8 (( Nc (x × { ∼ }) NC (x × { ∼ }) Nc (x × { ∼ })) → 1(x × { ∼ }) Tc Nc (x × { ∼ }))
4624, 36, 45mp2an 653 . . . . . . 7 1(x × { ∼ }) Tc Nc (x × { ∼ })
47 pw1eltc 6162 . . . . . . . 8 (( Nc (y × {}) NC (y × {}) Nc (y × {})) → 1(y × {}) Tc Nc (y × {}))
4825, 37, 47mp2an 653 . . . . . . 7 1(y × {}) Tc Nc (y × {})
49 pw1eq 4143 . . . . . . . . 9 (((x × { ∼ }) ∩ (y × {})) = 1((x × { ∼ }) ∩ (y × {})) = 1)
5039, 49ax-mp 8 . . . . . . . 8 1((x × { ∼ }) ∩ (y × {})) = 1
51 pw1in 4164 . . . . . . . 8 1((x × { ∼ }) ∩ (y × {})) = (1(x × { ∼ }) ∩ 1(y × {}))
52 pw10 4161 . . . . . . . 8 1 =
5350, 51, 523eqtr3i 2381 . . . . . . 7 (1(x × { ∼ }) ∩ 1(y × {})) =
54 eladdci 4399 . . . . . . 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 × {})))
5546, 48, 53, 54mp3an 1277 . . . . . 6 (1(x × { ∼ }) ∪ 1(y × {})) ( Tc Nc (x × { ∼ }) +c Tc Nc (y × {}))
5644, 55eqeltri 2423 . . . . 5 1((x × { ∼ }) ∪ (y × {})) ( Tc Nc (x × { ∼ }) +c Tc Nc (y × {}))
57 nceleq 6149 . . . . 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 × {})))
5829, 35, 43, 56, 57mp4an 654 . . . 4 Tc ( Nc (x × { ∼ }) +c Nc (y × {})) = ( Tc Nc (x × { ∼ }) +c Tc Nc (y × {}))
59 addceq12 4385 . . . . 5 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → (A +c B) = ( Nc (x × { ∼ }) +c Nc (y × {})))
60 tceq 6158 . . . . 5 ((A +c B) = ( Nc (x × { ∼ }) +c Nc (y × {})) → Tc (A +c B) = Tc ( Nc (x × { ∼ }) +c Nc (y × {})))
6159, 60syl 15 . . . 4 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → Tc (A +c B) = Tc ( Nc (x × { ∼ }) +c Nc (y × {})))
62 tceq 6158 . . . . . 6 (A = Nc (x × { ∼ }) → Tc A = Tc Nc (x × { ∼ }))
6362adantr 451 . . . . 5 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → Tc A = Tc Nc (x × { ∼ }))
64 tceq 6158 . . . . . 6 (B = Nc (y × {}) → Tc B = Tc Nc (y × {}))
6564adantl 452 . . . . 5 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → Tc B = Tc Nc (y × {}))
6663, 65addceq12d 4391 . . . 4 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → ( Tc A +c Tc B) = ( Tc Nc (x × { ∼ }) +c Tc Nc (y × {})))
6758, 61, 663eqtr4a 2411 . . 3 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → Tc (A +c B) = ( Tc A +c Tc B))
6867exlimivv 1635 . 2 (xy(A = Nc (x × { ∼ }) B = Nc (y × {})) → Tc (A +c B) = ( Tc A +c Tc B))
6923, 68sylbi 187 1 ((A NC B NC ) → Tc (A +c B) = ( Tc A +c Tc B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  ℘1cpw1 4135   +c cplc 4375   class class class wbr 4639   × cxp 4770   ≈ cen 6028   NC cncs 6088   Nc cnc 6091   Tc ctc 6093 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-tc 6103 This theorem is referenced by:  tc2c  6166  tlecg  6230  nmembers1  6271  nchoicelem1  6289  nchoicelem2  6290  nchoicelem17  6305
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