Theorem oe1m 4185

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 3975	. . 3 |- (x = (/) -> (1o ^o x) = (1o ^o (/)))
2	1	eqeq1d 1486	. 2 |- (x = (/) -> ((1o ^o x) = 1o <-> (1o ^o (/)) = 1o))
3		opreq2 3975	. . 3 |- (x = y -> (1o ^o x) = (1o ^o y))
4	3	eqeq1d 1486	. 2 |- (x = y -> ((1o ^o x) = 1o <-> (1o ^o y) = 1o))
5		opreq2 3975	. . 3 |- (x = suc y -> (1o ^o x) = (1o ^o suc y))
6	5	eqeq1d 1486	. 2 |- (x = suc y -> ((1o ^o x) = 1o <-> (1o ^o suc y) = 1o))
7		opreq2 3975	. . 3 |- (x = A -> (1o ^o x) = (1o ^o A))
8	7	eqeq1d 1486	. 2 |- (x = A -> ((1o ^o x) = 1o <-> (1o ^o A) = 1o))
9		1on 4144	. . 3 |- 1o e. On
10		oe0 4167	. . 3 |- (1o e. On -> (1o ^o (/)) = 1o)
11	9, 10	ax-mp 7	. 2 |- (1o ^o (/)) = 1o
12		oesuc 4172	. . . . 5 |- ((1o e. On /\ y e. On) -> (1o ^o suc y) = ((1o ^o y) .o 1o))
13	9, 12	mpan 697	. . . 4 |- (y e. On -> (1o ^o suc y) = ((1o ^o y) .o 1o))
14		opreq1 3974	. . . . 5 |- ((1o ^o y) = 1o -> ((1o ^o y) .o 1o) = (1o .o 1o))
15		om1 4182	. . . . . 6 |- (1o e. On -> (1o .o 1o) = 1o)
16	9, 15	ax-mp 7	. . . . 5 |- (1o .o 1o) = 1o
17	14, 16	syl6eq 1526	. . . 4 |- ((1o ^o y) = 1o -> ((1o ^o y) .o 1o) = 1o)
18	13, 17	sylan9eq 1530	. . 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 1816	. . . . . 6 |- x e. V
21		0lt1o 4153	. . . . . . . 8 |- (/) e. 1o
22		oelim 4175	. . . . . . . 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 698	. . . . . . 7 |- ((1o e. On /\ (x e. V /\ Lim x)) -> (1o ^o x) = U_y e. x (1o ^o y))
24	9, 23	mpan 697	. . . . . 6 |- ((x e. V /\ Lim x) -> (1o ^o x) = U_y e. x (1o ^o y))
25	20, 24	mpan 697	. . . . 5 |- (Lim x -> (1o ^o x) = U_y e. x (1o ^o y))
26	25	eqeq1d 1486	. . . 4 |- (Lim x -> ((1o ^o x) = 1o <-> U_y e. x (1o ^o y) = 1o))
27		0ellim 3037	. . . . . 6 |- (Lim x -> (/) e. x)
28		ne0i 2289	. . . . . 6 |- ((/) e. x -> x =/= (/))
29		iunconst 2576	. . . . . 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 1489	. . . 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 533	. . 3 |- (Lim x -> ((1o ^o x) = 1o <-> U_y e. x (1o ^o y) = U_y e. x 1o))
33		iuneq2 2582	. . 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 3167	1 |- (A e. On -> (1o ^o A) = 1o)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  Vcvv 1814  (/)c0 2283  U_ciun 2570  Oncon0 2954  Lim wlim 2955  suc csuc 2956  (class class class)co 3969  1oc1o 4134   .o comu 4137   ^o coe 4138

This theorem is referenced by:  oewordi 4224

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1o 4139  df-oadd 4141  df-omul 4142  df-oexp 4143

Copyright terms: Public domain