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Theorem tcdi 6164
Description: T raising distributes over addition. (Contributed by SF, 2-Mar-2015.)
Assertion
Ref Expression
tcdi NC NC Tc Tc Tc

Proof of Theorem tcdi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eeanv 1913 . . 3 Nc Nc Nc Nc
2 vex 2862 . . . . . . . 8
3 0ex 4110 . . . . . . . . 9
43complex 4104 . . . . . . . 8
52, 4xpsnen 6049 . . . . . . 7
6 snex 4111 . . . . . . . . 9
72, 6xpex 5115 . . . . . . . 8
87eqnc 6127 . . . . . . 7 Nc Nc
95, 8mpbir 200 . . . . . 6 Nc Nc
109eqeq2i 2363 . . . . 5 Nc Nc
11 vex 2862 . . . . . . . 8
1211, 3xpsnen 6049 . . . . . . 7
13 snex 4111 . . . . . . . . 9
1411, 13xpex 5115 . . . . . . . 8
1514eqnc 6127 . . . . . . 7 Nc Nc
1612, 15mpbir 200 . . . . . 6 Nc Nc
1716eqeq2i 2363 . . . . 5 Nc Nc
1810, 17anbi12i 678 . . . 4 Nc Nc Nc Nc
19182exbii 1583 . . 3 Nc Nc Nc Nc
20 elncs 6119 . . . 4 NC Nc
21 elncs 6119 . . . 4 NC Nc
2220, 21anbi12i 678 . . 3 NC NC Nc Nc
231, 19, 223bitr4ri 269 . 2 NC NC Nc Nc
247ncelncsi 6121 . . . . . . 7 Nc NC
2514ncelncsi 6121 . . . . . . 7 Nc NC
26 ncaddccl 6144 . . . . . . 7 Nc NC Nc NC Nc Nc NC
2724, 25, 26mp2an 653 . . . . . 6 Nc Nc NC
28 tccl 6160 . . . . . 6 Nc Nc NC Tc Nc Nc NC
2927, 28ax-mp 8 . . . . 5 Tc Nc Nc NC
30 tccl 6160 . . . . . . 7 Nc NC Tc Nc NC
3124, 30ax-mp 8 . . . . . 6 Tc Nc NC
32 tccl 6160 . . . . . . 7 Nc NC Tc Nc NC
3325, 32ax-mp 8 . . . . . 6 Tc Nc NC
34 ncaddccl 6144 . . . . . 6 Tc Nc NC Tc Nc NC Tc Nc Tc Nc NC
3531, 33, 34mp2an 653 . . . . 5 Tc Nc Tc Nc NC
367ncid 6123 . . . . . . 7 Nc
3714ncid 6123 . . . . . . 7 Nc
38 necompl 3544 . . . . . . . 8
394, 38xpnedisj 5513 . . . . . . 7
40 eladdci 4399 . . . . . . 7 Nc Nc Nc Nc
4136, 37, 39, 40mp3an 1277 . . . . . 6 Nc Nc
42 pw1eltc 6162 . . . . . 6 Nc Nc NC Nc Nc 1 Tc Nc Nc
4327, 41, 42mp2an 653 . . . . 5 1 Tc Nc Nc
44 pw1un 4163 . . . . . 6 1 1 1
45 pw1eltc 6162 . . . . . . . 8 Nc NC Nc 1 Tc Nc
4624, 36, 45mp2an 653 . . . . . . 7 1 Tc Nc
47 pw1eltc 6162 . . . . . . . 8 Nc NC Nc 1 Tc Nc
4825, 37, 47mp2an 653 . . . . . . 7 1 Tc Nc
49 pw1eq 4143 . . . . . . . . 9 1 1
5039, 49ax-mp 8 . . . . . . . 8 1 1
51 pw1in 4164 . . . . . . . 8 1 1 1
52 pw10 4161 . . . . . . . 8 1
5350, 51, 523eqtr3i 2381 . . . . . . 7 1 1
54 eladdci 4399 . . . . . . 7 1 Tc Nc 1 Tc Nc 1 1 1 1 Tc Nc Tc Nc
5546, 48, 53, 54mp3an 1277 . . . . . 6 1 1 Tc Nc Tc Nc
5644, 55eqeltri 2423 . . . . 5 1 Tc Nc Tc Nc
57 nceleq 6149 . . . . 5 Tc Nc Nc NC Tc Nc Tc Nc NC 1 Tc Nc Nc 1 Tc Nc Tc Nc Tc Nc Nc Tc Nc Tc Nc
5829, 35, 43, 56, 57mp4an 654 . . . 4 Tc Nc Nc Tc Nc Tc Nc
59 addceq12 4385 . . . . 5 Nc Nc Nc Nc
60 tceq 6158 . . . . 5 Nc Nc Tc Tc Nc Nc
6159, 60syl 15 . . . 4 Nc Nc Tc Tc Nc Nc
62 tceq 6158 . . . . . 6 Nc Tc Tc Nc
6362adantr 451 . . . . 5 Nc Nc Tc Tc Nc
64 tceq 6158 . . . . . 6 Nc Tc Tc Nc
6564adantl 452 . . . . 5 Nc Nc Tc Tc Nc
6663, 65addceq12d 4391 . . . 4 Nc Nc Tc Tc Tc Nc Tc Nc
6758, 61, 663eqtr4a 2411 . . 3 Nc Nc Tc Tc Tc
6867exlimivv 1635 . 2 Nc Nc Tc Tc Tc
6923, 68sylbi 187 1 NC NC Tc Tc Tc
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  1 cpw1 4135   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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