Theorem tce2 6237

Description: Distributive law for T-raising and cardinal exponentiation to two. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
tce2	|- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Tc (2c ↑c M) = (2c ↑c Tc M))

Proof of Theorem tce2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ce0ncpw1 6186	. 2 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> E.x M = Nc ~P1 x)
2		vex 2863	. . . . . . . 8 |- x e. _V
3	2	pwex 4330	. . . . . . 7 |- ~Px e. _V
4	3	tcnc 6226	. . . . . 6 |- Tc Nc ~Px = Nc ~P1 ~Px
5		eqid 2353	. . . . . . . 8 |- Nc ~P1 x = Nc ~P1 x
6	2	ce2 6193	. . . . . . . 8 |- ( Nc ~P1 x = Nc ~P1 x -> (2c ↑c Nc ~P1 x) = Nc ~Px)
7	5, 6	ax-mp 5	. . . . . . 7 |- (2c ↑c Nc ~P1 x) = Nc ~Px
8		tceq 6159	. . . . . . 7 |- ((2c ↑c Nc ~P1 x) = Nc ~Px -> Tc (2c ↑c Nc ~P1 x) = Tc Nc ~Px)
9	7, 8	ax-mp 5	. . . . . 6 |- Tc (2c ↑c Nc ~P1 x) = Tc Nc ~Px
10	2	ncpwpw1 6154	. . . . . 6 |- Nc ~P~P1 x = Nc ~P1 ~Px
11	4, 9, 10	3eqtr4i 2383	. . . . 5 |- Tc (2c ↑c Nc ~P1 x) = Nc ~P~P1 x
12		eqid 2353	. . . . . 6 |- Nc ~P1 ~P1 x = Nc ~P1 ~P1 x
13	2	pw1ex 4304	. . . . . . 7 |- ~P1 x e. _V
14	13	ce2 6193	. . . . . 6 |- ( Nc ~P1 ~P1 x = Nc ~P1 ~P1 x -> (2c ↑c Nc ~P1 ~P1 x) = Nc ~P~P1 x)
15	12, 14	ax-mp 5	. . . . 5 |- (2c ↑c Nc ~P1 ~P1 x) = Nc ~P~P1 x
16	11, 15	eqtr4i 2376	. . . 4 |- Tc (2c ↑c Nc ~P1 x) = (2c ↑c Nc ~P1 ~P1 x)
17		oveq2 5532	. . . . 5 |- (M = Nc ~P1 x -> (2c ↑c M) = (2c ↑c Nc ~P1 x))
18		tceq 6159	. . . . 5 |- ((2c ↑c M) = (2c ↑c Nc ~P1 x) -> Tc (2c ↑c M) = Tc (2c ↑c Nc ~P1 x))
19	17, 18	syl 15	. . . 4 |- (M = Nc ~P1 x -> Tc (2c ↑c M) = Tc (2c ↑c Nc ~P1 x))
20		tceq 6159	. . . . . 6 |- (M = Nc ~P1 x -> Tc M = Tc Nc ~P1 x)
21	13	tcnc 6226	. . . . . 6 |- Tc Nc ~P1 x = Nc ~P1 ~P1 x
22	20, 21	syl6eq 2401	. . . . 5 |- (M = Nc ~P1 x -> Tc M = Nc ~P1 ~P1 x)
23	22	oveq2d 5539	. . . 4 |- (M = Nc ~P1 x -> (2c ↑c Tc M) = (2c ↑c Nc ~P1 ~P1 x))
24	16, 19, 23	3eqtr4a 2411	. . 3 |- (M = Nc ~P1 x -> Tc (2c ↑c M) = (2c ↑c Tc M))
25	24	exlimiv 1634	. 2 |- (E.x M = Nc ~P1 x -> Tc (2c ↑c M) = (2c ↑c Tc M))
26	1, 25	syl 15	1 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Tc (2c ↑c M) = (2c ↑c Tc M))

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  ~Pcpw 3723  ~P₁ cpw1 4136  0_cc0c 4375  (class class class)co 5526   NC cncs 6089   Nc cnc 6092   T_c ctc 6094  2_cc2c 6095   ↑_c cce 6097

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-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-tc 6104  df-2c 6105  df-ce 6107

This theorem is referenced by:  nchoicelem12  6301  nchoicelem17  6306