Theorem oe1m 7625

Description: Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)

Assertion

Ref	Expression
oe1m	|- ( A e. On -> ( 1o ^o A ) = 1o )

Proof of Theorem oe1m

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 |- ( x = (/) -> ( 1o ^o x ) = ( 1o ^o (/) ) )
2	1	eqeq1d 2624	. 2 |- ( x = (/) -> ( ( 1o ^o x ) = 1o <-> ( 1o ^o (/) ) = 1o ) )
3		oveq2 6658	. . 3 |- ( x = y -> ( 1o ^o x ) = ( 1o ^o y ) )
4	3	eqeq1d 2624	. 2 |- ( x = y -> ( ( 1o ^o x ) = 1o <-> ( 1o ^o y ) = 1o ) )
5		oveq2 6658	. . 3 |- ( x = suc y -> ( 1o ^o x ) = ( 1o ^o suc y ) )
6	5	eqeq1d 2624	. 2 |- ( x = suc y -> ( ( 1o ^o x ) = 1o <-> ( 1o ^o suc y ) = 1o ) )
7		oveq2 6658	. . 3 |- ( x = A -> ( 1o ^o x ) = ( 1o ^o A ) )
8	7	eqeq1d 2624	. 2 |- ( x = A -> ( ( 1o ^o x ) = 1o <-> ( 1o ^o A ) = 1o ) )
9		1on 7567	. . 3 |- 1o e. On
10		oe0 7602	. . 3 |- ( 1o e. On -> ( 1o ^o (/) ) = 1o )
11	9, 10	ax-mp 5	. 2 |- ( 1o ^o (/) ) = 1o
12		oesuc 7607	. . . . 5 |- ( ( 1o e. On /\ y e. On ) -> ( 1o ^o suc y ) = ( ( 1o ^o y ) .o 1o ) )
13	9, 12	mpan 706	. . . 4 |- ( y e. On -> ( 1o ^o suc y ) = ( ( 1o ^o y ) .o 1o ) )
14		oveq1 6657	. . . . 5 |- ( ( 1o ^o y ) = 1o -> ( ( 1o ^o y ) .o 1o ) = ( 1o .o 1o ) )
15		om1 7622	. . . . . 6 |- ( 1o e. On -> ( 1o .o 1o ) = 1o )
16	9, 15	ax-mp 5	. . . . 5 |- ( 1o .o 1o ) = 1o
17	14, 16	syl6eq 2672	. . . 4 |- ( ( 1o ^o y ) = 1o -> ( ( 1o ^o y ) .o 1o ) = 1o )
18	13, 17	sylan9eq 2676	. . 3 |- ( ( y e. On /\ ( 1o ^o y ) = 1o ) -> ( 1o ^o suc y ) = 1o )
19	18	ex 450	. 2 |- ( y e. On -> ( ( 1o ^o y ) = 1o -> ( 1o ^o suc y ) = 1o ) )
20		iuneq2 4537	. . 3 |- ( A. y e. x ( 1o ^o y ) = 1o -> U_ y e. x ( 1o ^o y ) = U_ y e. x 1o )
21		vex 3203	. . . . . 6 |- x e. _V
22		0lt1o 7584	. . . . . . . 8 |- (/) e. 1o
23		oelim 7614	. . . . . . . 8 |- ( ( ( 1o e. On /\ ( x e. _V /\ Lim x ) ) /\ (/) e. 1o ) -> ( 1o ^o x ) = U_ y e. x ( 1o ^o y ) )
24	22, 23	mpan2 707	. . . . . . 7 |- ( ( 1o e. On /\ ( x e. _V /\ Lim x ) ) -> ( 1o ^o x ) = U_ y e. x ( 1o ^o y ) )
25	9, 24	mpan 706	. . . . . 6 |- ( ( x e. _V /\ Lim x ) -> ( 1o ^o x ) = U_ y e. x ( 1o ^o y ) )
26	21, 25	mpan 706	. . . . 5 |- ( Lim x -> ( 1o ^o x ) = U_ y e. x ( 1o ^o y ) )
27	26	eqeq1d 2624	. . . 4 |- ( Lim x -> ( ( 1o ^o x ) = 1o <-> U_ y e. x ( 1o ^o y ) = 1o ) )
28		0ellim 5787	. . . . . 6 |- ( Lim x -> (/) e. x )
29		ne0i 3921	. . . . . 6 |- ( (/) e. x -> x =/= (/) )
30		iunconst 4529	. . . . . 6 |- ( x =/= (/) -> U_ y e. x 1o = 1o )
31	28, 29, 30	3syl 18	. . . . 5 |- ( Lim x -> U_ y e. x 1o = 1o )
32	31	eqeq2d 2632	. . . 4 |- ( Lim x -> ( U_ y e. x ( 1o ^o y ) = U_ y e. x 1o <-> U_ y e. x ( 1o ^o y ) = 1o ) )
33	27, 32	bitr4d 271	. . 3 |- ( Lim x -> ( ( 1o ^o x ) = 1o <-> U_ y e. x ( 1o ^o y ) = U_ y e. x 1o ) )
34	20, 33	syl5ibr 236	. 2 |- ( Lim x -> ( A. y e. x ( 1o ^o y ) = 1o -> ( 1o ^o x ) = 1o ) )
35	2, 4, 6, 8, 11, 19, 34	tfinds 7059	1 |- ( A e. On -> ( 1o ^o A ) = 1o )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   U_ciun 4520   Oncon0 5723   Lim wlim 5724   suc csuc 5725  (class class class)co 6650   1oc1o 7553   .o comu 7558   ^o coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oewordi  7671  oeoe  7679  cantnflem2  8587