Theorem oe1m 4163

Description: Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67.

Assertion

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

Proof of Theorem oe1m

Step	Hyp	Ref	Expression
1		opreq2 3954	. . 3 |- (x = (/) -> (1o ^o x) = (1o ^o (/)))
2	1	eqeq1d 1475	. 2 |- (x = (/) -> ((1o ^o x) = 1o <-> (1o ^o (/)) = 1o))
3		opreq2 3954	. . 3 |- (x = y -> (1o ^o x) = (1o ^o y))
4	3	eqeq1d 1475	. 2 |- (x = y -> ((1o ^o x) = 1o <-> (1o ^o y) = 1o))
5		opreq2 3954	. . 3 |- (x = suc y -> (1o ^o x) = (1o ^o suc y))
6	5	eqeq1d 1475	. 2 |- (x = suc y -> ((1o ^o x) = 1o <-> (1o ^o suc y) = 1o))
7		opreq2 3954	. . 3 |- (x = A -> (1o ^o x) = (1o ^o A))
8	7	eqeq1d 1475	. 2 |- (x = A -> ((1o ^o x) = 1o <-> (1o ^o A) = 1o))
9		1on 4122	. . 3 |- 1o e. On
10		oe0 4145	. . 3 |- (1o e. On -> (1o ^o (/)) = 1o)
11	9, 10	ax-mp 7	. 2 |- (1o ^o (/)) = 1o
12		oesuc 4150	. . . . 5 |- ((1o e. On /\ y e. On) -> (1o ^o suc y) = ((1o ^o y) .o 1o))
13	9, 12	mpan 693	. . . 4 |- (y e. On -> (1o ^o suc y) = ((1o ^o y) .o 1o))
14		opreq1 3953	. . . . 5 |- ((1o ^o y) = 1o -> ((1o ^o y) .o 1o) = (1o .o 1o))
15		om1 4160	. . . . . 6 |- (1o e. On -> (1o .o 1o) = 1o)
16	9, 15	ax-mp 7	. . . . 5 |- (1o .o 1o) = 1o
17	14, 16	syl6eq 1515	. . . 4 |- ((1o ^o y) = 1o -> ((1o ^o y) .o 1o) = 1o)
18	13, 17	sylan9eq 1519	. . 3 |- ((y e. On /\ (1o ^o y) = 1o) -> (1o ^o suc y) = 1o)
19	18	ex 373	. 2 |- (y e. On -> ((1o ^o y) = 1o -> (1o ^o suc y) = 1o))
20		visset 1804	. . . . . 6 |- x e. V
21		0lt1o 4131	. . . . . . . 8 |- (/) e. 1o
22		oelim 4153	. . . . . . . 8 |- (((1o e. On /\ (x e. V /\ Lim x)) /\ (/) e. 1o) -> (1o ^o x) = U_y e. x (1o ^o y))
23	21, 22	mpan2 694	. . . . . . 7 |- ((1o e. On /\ (x e. V /\ Lim x)) -> (1o ^o x) = U_y e. x (1o ^o y))
24	9, 23	mpan 693	. . . . . 6 |- ((x e. V /\ Lim x) -> (1o ^o x) = U_y e. x (1o ^o y))
25	20, 24	mpan 693	. . . . 5 |- (Lim x -> (1o ^o x) = U_y e. x (1o ^o y))
26	25	eqeq1d 1475	. . . 4 |- (Lim x -> ((1o ^o x) = 1o <-> U_y e. x (1o ^o y) = 1o))
27		0ellim 3021	. . . . . 6 |- (Lim x -> (/) e. x)
28		ne0i 2276	. . . . . 6 |- ((/) e. x -> x =/= (/))
29		iunconst 2562	. . . . . 6 |- (x =/= (/) -> U_y e. x 1o = 1o)
30	27, 28, 29	3syl 20	. . . . 5 |- (Lim x -> U_y e. x 1o = 1o)
31	30	eqeq2d 1478	. . . 4 |- (Lim x -> (U_y e. x (1o ^o y) = U_y e. x 1o <-> U_y e. x (1o ^o y) = 1o))
32	26, 31	bitr4d 529	. . 3 |- (Lim x -> ((1o ^o x) = 1o <-> U_y e. x (1o ^o y) = U_y e. x 1o))
33		iuneq2 2568	. . 3 |- (A.y e. x (1o ^o y) = 1o -> U_y e. x (1o ^o y) = U_y e. x 1o)
34	32, 33	syl5bir 210	. 2 |- (Lim x -> (A.y e. x (1o ^o y) = 1o -> (1o ^o x) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 3151	1 |- (A e. On -> (1o ^o A) = 1o)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  Vcvv 1802  (/)c0 2270  U_ciun 2556  Oncon0 2938  Lim wlim 2939  suc csuc 2940  (class class class)co 3948  1oc1o 4112   .o comu 4115   ^o coe 4116

This theorem is referenced by:  oewordi 4202

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1o 4117  df-oadd 4119  df-omul 4120  df-oexp 4121

Copyright terms: Public domain