Theorem tfin1c 4500

Description: The finite T operator is idempotent over 1_c. Theorem X.1.34(a) of [Rosser] p. 529. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
tfin1c	|- Tfin 1c = 1c

Proof of Theorem tfin1c

Step	Hyp	Ref	Expression
1		peano1 4403	. . 3 |- 0c e. Nn
2		addcid2 4408	. . . 4 |- (0c +o 1c) = 1c
3		1cex 4143	. . . . . 6 |- 1c e. _V
4	3	snel1c 4141	. . . . 5 |- {1c} e. 1c
5		ne0i 3557	. . . . 5 |- ({1c} e. 1c -> 1c =/= (/))
6	4, 5	ax-mp 5	. . . 4 |- 1c =/= (/)
7	2, 6	eqnetri 2534	. . 3 |- (0c +o 1c) =/= (/)
8		tfinsuc 4499	. . 3 |- ((0c e. Nn /\ (0c +o 1c) =/= (/)) -> Tfin (0c +o 1c) = ( Tfin 0c +o 1c))
9	1, 7, 8	mp2an 653	. 2 |- Tfin (0c +o 1c) = ( Tfin 0c +o 1c)
10		tfineq 4489	. . 3 |- ((0c +o 1c) = 1c -> Tfin (0c +o 1c) = Tfin 1c)
11	2, 10	ax-mp 5	. 2 |- Tfin (0c +o 1c) = Tfin 1c
12		tfin0c 4498	. . . 4 |- Tfin 0c = 0c
13	12	addceq1i 4387	. . 3 |- ( Tfin 0c +o 1c) = (0c +o 1c)
14	13, 2	eqtri 2373	. 2 |- ( Tfin 0c +o 1c) = 1c
15	9, 11, 14	3eqtr3i 2381	1 |- Tfin 1c = 1c

Syntax hints:   = wceq 1642   e. wcel 1710   =/= wne 2517  (/)c0 3551  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376   T_fin ctfin 4436

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-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-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-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-tfin 4444

This theorem is referenced by:  oddtfin  4519  sfintfin  4533