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 ∧ (M ↑c 0c) ∈ 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 6185	. 2 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ∃x M = Nc ℘1x)
2		vex 2862	. . . . . . . 8 ⊢ x ∈ V
3	2	pwex 4329	. . . . . . 7 ⊢ ℘x ∈ V
4	3	tcnc 6225	. . . . . 6 ⊢ Tc Nc ℘x = Nc ℘1℘x
5		eqid 2353	. . . . . . . 8 ⊢ Nc ℘1x = Nc ℘1x
6	2	ce2 6192	. . . . . . . 8 ⊢ ( Nc ℘1x = Nc ℘1x → (2c ↑c Nc ℘1x) = Nc ℘x)
7	5, 6	ax-mp 8	. . . . . . 7 ⊢ (2c ↑c Nc ℘1x) = Nc ℘x
8		tceq 6158	. . . . . . 7 ⊢ ((2c ↑c Nc ℘1x) = Nc ℘x → Tc (2c ↑c Nc ℘1x) = Tc Nc ℘x)
9	7, 8	ax-mp 8	. . . . . 6 ⊢ Tc (2c ↑c Nc ℘1x) = Tc Nc ℘x
10	2	ncpwpw1 6153	. . . . . 6 ⊢ Nc ℘℘1x = Nc ℘1℘x
11	4, 9, 10	3eqtr4i 2383	. . . . 5 ⊢ Tc (2c ↑c Nc ℘1x) = Nc ℘℘1x
12		eqid 2353	. . . . . 6 ⊢ Nc ℘1℘1x = Nc ℘1℘1x
13	2	pw1ex 4303	. . . . . . 7 ⊢ ℘1x ∈ V
14	13	ce2 6192	. . . . . 6 ⊢ ( Nc ℘1℘1x = Nc ℘1℘1x → (2c ↑c Nc ℘1℘1x) = Nc ℘℘1x)
15	12, 14	ax-mp 8	. . . . 5 ⊢ (2c ↑c Nc ℘1℘1x) = Nc ℘℘1x
16	11, 15	eqtr4i 2376	. . . 4 ⊢ Tc (2c ↑c Nc ℘1x) = (2c ↑c Nc ℘1℘1x)
17		oveq2 5531	. . . . 5 ⊢ (M = Nc ℘1x → (2c ↑c M) = (2c ↑c Nc ℘1x))
18		tceq 6158	. . . . 5 ⊢ ((2c ↑c M) = (2c ↑c Nc ℘1x) → Tc (2c ↑c M) = Tc (2c ↑c Nc ℘1x))
19	17, 18	syl 15	. . . 4 ⊢ (M = Nc ℘1x → Tc (2c ↑c M) = Tc (2c ↑c Nc ℘1x))
20		tceq 6158	. . . . . 6 ⊢ (M = Nc ℘1x → Tc M = Tc Nc ℘1x)
21	13	tcnc 6225	. . . . . 6 ⊢ Tc Nc ℘1x = Nc ℘1℘1x
22	20, 21	syl6eq 2401	. . . . 5 ⊢ (M = Nc ℘1x → Tc M = Nc ℘1℘1x)
23	22	oveq2d 5538	. . . 4 ⊢ (M = Nc ℘1x → (2c ↑c Tc M) = (2c ↑c Nc ℘1℘1x))
24	16, 19, 23	3eqtr4a 2411	. . 3 ⊢ (M = Nc ℘1x → Tc (2c ↑c M) = (2c ↑c Tc M))
25	24	exlimiv 1634	. 2 ⊢ (∃x M = Nc ℘1x → Tc (2c ↑c M) = (2c ↑c Tc M))
26	1, 25	syl 15	1 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Tc (2c ↑c M) = (2c ↑c Tc M))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ℘cpw 3722  ℘₁cpw1 4135  0_cc0c 4374  (class class class)co 5525   NC cncs 6088   Nc cnc 6091   T_c ctc 6093  2_cc2c 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