Theorem oe1m 5226

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 4890	. . 3 |- (x = (/) -> (1o ^o x) = (1o ^o (/)))
2	1	eqeq1d 1892	. 2 |- (x = (/) -> ((1o ^o x) = 1o <-> (1o ^o (/)) = 1o))
3		opreq2 4890	. . 3 |- (x = y -> (1o ^o x) = (1o ^o y))
4	3	eqeq1d 1892	. 2 |- (x = y -> ((1o ^o x) = 1o <-> (1o ^o y) = 1o))
5		opreq2 4890	. . 3 |- (x = suc y -> (1o ^o x) = (1o ^o suc y))
6	5	eqeq1d 1892	. 2 |- (x = suc y -> ((1o ^o x) = 1o <-> (1o ^o suc y) = 1o))
7		opreq2 4890	. . 3 |- (x = A -> (1o ^o x) = (1o ^o A))
8	7	eqeq1d 1892	. 2 |- (x = A -> ((1o ^o x) = 1o <-> (1o ^o A) = 1o))
9		1on 5182	. . 3 |- 1o e. On
10		oe0 5206	. . 3 |- (1o e. On -> (1o ^o (/)) = 1o)
11	9, 10	ax-mp 7	. 2 |- (1o ^o (/)) = 1o
12		oesuc 5211	. . . . 5 |- ((1o e. On /\ y e. On) -> (1o ^o suc y) = ((1o ^o y) .o 1o))
13	9, 12	mpan 759	. . . 4 |- (y e. On -> (1o ^o suc y) = ((1o ^o y) .o 1o))
14		opreq1 4889	. . . . 5 |- ((1o ^o y) = 1o -> ((1o ^o y) .o 1o) = (1o .o 1o))
15		om1 5223	. . . . . 6 |- (1o e. On -> (1o .o 1o) = 1o)
16	9, 15	ax-mp 7	. . . . 5 |- (1o .o 1o) = 1o
17	14, 16	syl6eq 1944	. . . 4 |- ((1o ^o y) = 1o -> ((1o ^o y) .o 1o) = 1o)
18	13, 17	sylan9eq 1948	. . 3 |- ((y e. On /\ (1o ^o y) = 1o) -> (1o ^o suc y) = 1o)
19	18	ex 402	. 2 |- (y e. On -> ((1o ^o y) = 1o -> (1o ^o suc y) = 1o))
20		visset 2295	. . . . . 6 |- x e. _V
21		0lt1o 5192	. . . . . . . 8 |- (/) e. 1o
22		oelim 5214	. . . . . . . 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 760	. . . . . . 7 |- ((1o e. On /\ (x e. _V /\ Lim x)) -> (1o ^o x) = U_y e. x (1o ^o y))
24	9, 23	mpan 759	. . . . . 6 |- ((x e. _V /\ Lim x) -> (1o ^o x) = U_y e. x (1o ^o y))
25	20, 24	mpan 759	. . . . 5 |- (Lim x -> (1o ^o x) = U_y e. x (1o ^o y))
26	25	eqeq1d 1892	. . . 4 |- (Lim x -> ((1o ^o x) = 1o <-> U_y e. x (1o ^o y) = 1o))
27		0ellim 3726	. . . . . 6 |- (Lim x -> (/) e. x)
28		ne0i 2881	. . . . . 6 |- ((/) e. x -> x =/= (/))
29		iunconst 3262	. . . . . 6 |- (x =/= (/) -> U_y e. x 1o = 1o)
30	27, 28, 29	3syl 24	. . . . 5 |- (Lim x -> U_y e. x 1o = 1o)
31	30	eqeq2d 1895	. . . 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 590	. . 3 |- (Lim x -> ((1o ^o x) = 1o <-> U_y e. x (1o ^o y) = U_y e. x 1o))
33		iuneq2 3273	. . 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 227	. 2 |- (Lim x -> (A.y e. x (1o ^o y) = 1o -> (1o ^o x) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 3942	1 |- (A e. On -> (1o ^o A) = 1o)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292  (/)c0 2875  U_ciun 3255  Oncon0 3657  Lim wlim 3658  suc csuc 3659  (class class class)co 4884  1oc1o 5172   .o comu 5175   ^o coe 5176

This theorem is referenced by:  oewordi 5266  oeoe 5274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-oexp 5181

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