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Theorem tce2 6236
 Description: Distributive law for T-raising and cardinal exponentiation to two. (Contributed by SF, 13-Mar-2015.)
Assertion
Ref Expression
tce2 ((M NC (Mc 0c) NC ) → Tc (2cc M) = (2cc Tc M))

Proof of Theorem tce2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ce0ncpw1 6185 . 2 ((M NC (Mc 0c) NC ) → x M = Nc 1x)
2 vex 2862 . . . . . . . 8 x V
32pwex 4329 . . . . . . 7 x V
43tcnc 6225 . . . . . 6 Tc Nc x = Nc 1x
5 eqid 2353 . . . . . . . 8 Nc 1x = Nc 1x
62ce2 6192 . . . . . . . 8 ( Nc 1x = Nc 1x → (2cc Nc 1x) = Nc x)
75, 6ax-mp 8 . . . . . . 7 (2cc Nc 1x) = Nc x
8 tceq 6158 . . . . . . 7 ((2cc Nc 1x) = Nc xTc (2cc Nc 1x) = Tc Nc x)
97, 8ax-mp 8 . . . . . 6 Tc (2cc Nc 1x) = Tc Nc x
102ncpwpw1 6153 . . . . . 6 Nc 1x = Nc 1x
114, 9, 103eqtr4i 2383 . . . . 5 Tc (2cc Nc 1x) = Nc 1x
12 eqid 2353 . . . . . 6 Nc 11x = Nc 11x
132pw1ex 4303 . . . . . . 7 1x V
1413ce2 6192 . . . . . 6 ( Nc 11x = Nc 11x → (2cc Nc 11x) = Nc 1x)
1512, 14ax-mp 8 . . . . 5 (2cc Nc 11x) = Nc 1x
1611, 15eqtr4i 2376 . . . 4 Tc (2cc Nc 1x) = (2cc Nc 11x)
17 oveq2 5531 . . . . 5 (M = Nc 1x → (2cc M) = (2cc Nc 1x))
18 tceq 6158 . . . . 5 ((2cc M) = (2cc Nc 1x) → Tc (2cc M) = Tc (2cc Nc 1x))
1917, 18syl 15 . . . 4 (M = Nc 1xTc (2cc M) = Tc (2cc Nc 1x))
20 tceq 6158 . . . . . 6 (M = Nc 1xTc M = Tc Nc 1x)
2113tcnc 6225 . . . . . 6 Tc Nc 1x = Nc 11x
2220, 21syl6eq 2401 . . . . 5 (M = Nc 1xTc M = Nc 11x)
2322oveq2d 5538 . . . 4 (M = Nc 1x → (2cc Tc M) = (2cc Nc 11x))
2416, 19, 233eqtr4a 2411 . . 3 (M = Nc 1xTc (2cc M) = (2cc Tc M))
2524exlimiv 1634 . 2 (x M = Nc 1xTc (2cc M) = (2cc Tc M))
261, 25syl 15 1 ((M NC (Mc 0c) NC ) → Tc (2cc M) = (2cc Tc M))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ℘cpw 3722  ℘1cpw1 4135  0cc0c 4374  (class class class)co 5525   NC cncs 6088   Nc cnc 6091   Tc ctc 6093  2cc2c 6094   ↑c cce 6096 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-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-tc 6103  df-2c 6104  df-ce 6106 This theorem is referenced by:  nchoicelem12  6300  nchoicelem17  6305
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