Theorem oewordri 5267

Description: Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68.

Assertion

Ref	Expression
oewordri	|- ((B e. On /\ C e. On) -> (A e. B -> (A ^o C) C_ (B ^o C)))

Proof of Theorem oewordri

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (x = (/) -> (A ^o x) = (A ^o (/)))
2		opreq2 4890	. . . . 5 |- (x = (/) -> (B ^o x) = (B ^o (/)))
3	1, 2	sseq12d 2646	. . . 4 |- (x = (/) -> ((A ^o x) C_ (B ^o x) <-> (A ^o (/)) C_ (B ^o (/))))
4		opreq2 4890	. . . . 5 |- (x = y -> (A ^o x) = (A ^o y))
5		opreq2 4890	. . . . 5 |- (x = y -> (B ^o x) = (B ^o y))
6	4, 5	sseq12d 2646	. . . 4 |- (x = y -> ((A ^o x) C_ (B ^o x) <-> (A ^o y) C_ (B ^o y)))
7		opreq2 4890	. . . . 5 |- (x = suc y -> (A ^o x) = (A ^o suc y))
8		opreq2 4890	. . . . 5 |- (x = suc y -> (B ^o x) = (B ^o suc y))
9	7, 8	sseq12d 2646	. . . 4 |- (x = suc y -> ((A ^o x) C_ (B ^o x) <-> (A ^o suc y) C_ (B ^o suc y)))
10		opreq2 4890	. . . . 5 |- (x = C -> (A ^o x) = (A ^o C))
11		opreq2 4890	. . . . 5 |- (x = C -> (B ^o x) = (B ^o C))
12	10, 11	sseq12d 2646	. . . 4 |- (x = C -> ((A ^o x) C_ (B ^o x) <-> (A ^o C) C_ (B ^o C)))
13		onelon 3683	. . . . . . 7 |- ((B e. On /\ A e. B) -> A e. On)
14		oe0 5206	. . . . . . 7 |- (A e. On -> (A ^o (/)) = 1o)
15	13, 14	syl 12	. . . . . 6 |- ((B e. On /\ A e. B) -> (A ^o (/)) = 1o)
16		oe0 5206	. . . . . . 7 |- (B e. On -> (B ^o (/)) = 1o)
17	16	adantr 425	. . . . . 6 |- ((B e. On /\ A e. B) -> (B ^o (/)) = 1o)
18	15, 17	eqtr4d 1928	. . . . 5 |- ((B e. On /\ A e. B) -> (A ^o (/)) = (B ^o (/)))
19		eqimss 2665	. . . . 5 |- ((A ^o (/)) = (B ^o (/)) -> (A ^o (/)) C_ (B ^o (/)))
20	18, 19	syl 12	. . . 4 |- ((B e. On /\ A e. B) -> (A ^o (/)) C_ (B ^o (/)))
21		oecl 5218	. . . . . . . . . . . . . 14 |- ((A e. On /\ y e. On) -> (A ^o y) e. On)
22	21	3adant2 895	. . . . . . . . . . . . 13 |- ((A e. On /\ B e. On /\ y e. On) -> (A ^o y) e. On)
23		oecl 5218	. . . . . . . . . . . . . 14 |- ((B e. On /\ y e. On) -> (B ^o y) e. On)
24	23	3adant1 894	. . . . . . . . . . . . 13 |- ((A e. On /\ B e. On /\ y e. On) -> (B ^o y) e. On)
25		simp1 876	. . . . . . . . . . . . 13 |- ((A e. On /\ B e. On /\ y e. On) -> A e. On)
26		omwordri 5251	. . . . . . . . . . . . 13 |- (((A ^o y) e. On /\ (B ^o y) e. On /\ A e. On) -> ((A ^o y) C_ (B ^o y) -> ((A ^o y) .o A) C_ ((B ^o y) .o A)))
27	22, 24, 25, 26	syl111anc 1100	. . . . . . . . . . . 12 |- ((A e. On /\ B e. On /\ y e. On) -> ((A ^o y) C_ (B ^o y) -> ((A ^o y) .o A) C_ ((B ^o y) .o A)))
28	27	imp 377	. . . . . . . . . . 11 |- (((A e. On /\ B e. On /\ y e. On) /\ (A ^o y) C_ (B ^o y)) -> ((A ^o y) .o A) C_ ((B ^o y) .o A))
29	28	adantrl 430	. . . . . . . . . 10 |- (((A e. On /\ B e. On /\ y e. On) /\ (A C_ B /\ (A ^o y) C_ (B ^o y))) -> ((A ^o y) .o A) C_ ((B ^o y) .o A))
30		omwordi 5250	. . . . . . . . . . . . 13 |- ((A e. On /\ B e. On /\ (B ^o y) e. On) -> (A C_ B -> ((B ^o y) .o A) C_ ((B ^o y) .o B)))
31	30, 24	syld3an3 1142	. . . . . . . . . . . 12 |- ((A e. On /\ B e. On /\ y e. On) -> (A C_ B -> ((B ^o y) .o A) C_ ((B ^o y) .o B)))
32	31	imp 377	. . . . . . . . . . 11 |- (((A e. On /\ B e. On /\ y e. On) /\ A C_ B) -> ((B ^o y) .o A) C_ ((B ^o y) .o B))
33	32	adantrr 431	. . . . . . . . . 10 |- (((A e. On /\ B e. On /\ y e. On) /\ (A C_ B /\ (A ^o y) C_ (B ^o y))) -> ((B ^o y) .o A) C_ ((B ^o y) .o B))
34	29, 33	sstrd 2627	. . . . . . . . 9 |- (((A e. On /\ B e. On /\ y e. On) /\ (A C_ B /\ (A ^o y) C_ (B ^o y))) -> ((A ^o y) .o A) C_ ((B ^o y) .o B))
35		oesuc 5211	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
36	35	3adant2 895	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
37	36	adantr 425	. . . . . . . . 9 |- (((A e. On /\ B e. On /\ y e. On) /\ (A C_ B /\ (A ^o y) C_ (B ^o y))) -> (A ^o suc y) = ((A ^o y) .o A))
38		oesuc 5211	. . . . . . . . . . 11 |- ((B e. On /\ y e. On) -> (B ^o suc y) = ((B ^o y) .o B))
39	38	3adant1 894	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ y e. On) -> (B ^o suc y) = ((B ^o y) .o B))
40	39	adantr 425	. . . . . . . . 9 |- (((A e. On /\ B e. On /\ y e. On) /\ (A C_ B /\ (A ^o y) C_ (B ^o y))) -> (B ^o suc y) = ((B ^o y) .o B))
41	34, 37, 40	3sstr4d 2660	. . . . . . . 8 |- (((A e. On /\ B e. On /\ y e. On) /\ (A C_ B /\ (A ^o y) C_ (B ^o y))) -> (A ^o suc y) C_ (B ^o suc y))
42	41	exp520 1089	. . . . . . 7 |- (A e. On -> (B e. On -> (y e. On -> (A C_ B -> ((A ^o y) C_ (B ^o y) -> (A ^o suc y) C_ (B ^o suc y))))))
43	42	com3r 39	. . . . . 6 |- (y e. On -> (A e. On -> (B e. On -> (A C_ B -> ((A ^o y) C_ (B ^o y) -> (A ^o suc y) C_ (B ^o suc y))))))
44	43	imp4c 393	. . . . 5 |- (y e. On -> (((A e. On /\ B e. On) /\ A C_ B) -> ((A ^o y) C_ (B ^o y) -> (A ^o suc y) C_ (B ^o suc y))))
45		simpl 346	. . . . . 6 |- ((B e. On /\ A e. B) -> B e. On)
46		onelss 3705	. . . . . . 7 |- (B e. On -> (A e. B -> A C_ B))
47	46	imp 377	. . . . . 6 |- ((B e. On /\ A e. B) -> A C_ B)
48	13, 45, 47	jca31 311	. . . . 5 |- ((B e. On /\ A e. B) -> ((A e. On /\ B e. On) /\ A C_ B))
49	44, 48	syl5 20	. . . 4 |- (y e. On -> ((B e. On /\ A e. B) -> ((A ^o y) C_ (B ^o y) -> (A ^o suc y) C_ (B ^o suc y))))
50		opreq1 4889	. . . . . . . . . . 11 |- (A = (/) -> (A ^o x) = ((/) ^o x))
51	50	sseq1d 2644	. . . . . . . . . 10 |- (A = (/) -> ((A ^o x) C_ (B ^o x) <-> ((/) ^o x) C_ (B ^o x)))
52		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
53		limelon 3727	. . . . . . . . . . . . 13 |- ((x e. _V /\ Lim x) -> x e. On)
54	52, 53	mpan 759	. . . . . . . . . . . 12 |- (Lim x -> x e. On)
55		0ellim 3726	. . . . . . . . . . . 12 |- (Lim x -> (/) e. x)
56		oe0m1 5205	. . . . . . . . . . . . 13 |- (x e. On -> ((/) e. x <-> ((/) ^o x) = (/)))
57	56	biimpa 460	. . . . . . . . . . . 12 |- ((x e. On /\ (/) e. x) -> ((/) ^o x) = (/))
58	54, 55, 57	syl11anc 524	. . . . . . . . . . 11 |- (Lim x -> ((/) ^o x) = (/))
59		0ss 2900	. . . . . . . . . . . 12 |- (/) C_ (B ^o x)
60	59	a1i 8	. . . . . . . . . . 11 |- (Lim x -> (/) C_ (B ^o x))
61	58, 60	eqsstrd 2651	. . . . . . . . . 10 |- (Lim x -> ((/) ^o x) C_ (B ^o x))
62	51, 61	syl5bir 227	. . . . . . . . 9 |- (A = (/) -> (Lim x -> (A ^o x) C_ (B ^o x)))
63	62	adantl 424	. . . . . . . 8 |- (((B e. On /\ A e. B) /\ A = (/)) -> (Lim x -> (A ^o x) C_ (B ^o x)))
64	63	a1dd 53	. . . . . . 7 |- (((B e. On /\ A e. B) /\ A = (/)) -> (Lim x -> (A.y e. x (A ^o y) C_ (B ^o y) -> (A ^o x) C_ (B ^o x))))
65		oelim 5214	. . . . . . . . . . . . 13 |- (((A e. On /\ (x e. _V /\ Lim x)) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
66	52, 65	mpanlr1 779	. . . . . . . . . . . 12 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
67	66	an1rs 547	. . . . . . . . . . 11 |- (((A e. On /\ (/) e. A) /\ Lim x) -> (A ^o x) = U_y e. x (A ^o y))
68	67	adantllr 433	. . . . . . . . . 10 |- ((((A e. On /\ (B e. On /\ A e. B)) /\ (/) e. A) /\ Lim x) -> (A ^o x) = U_y e. x (A ^o y))
69	45	anim1i 361	. . . . . . . . . . . . 13 |- (((B e. On /\ A e. B) /\ Lim x) -> (B e. On /\ Lim x))
70		on0eln0 3718	. . . . . . . . . . . . . . . 16 |- (B e. On -> ((/) e. B <-> B =/= (/)))
71		ne0i 2881	. . . . . . . . . . . . . . . 16 |- (A e. B -> B =/= (/))
72	70, 71	syl5bir 227	. . . . . . . . . . . . . . 15 |- (B e. On -> (A e. B -> (/) e. B))
73	72	imp 377	. . . . . . . . . . . . . 14 |- ((B e. On /\ A e. B) -> (/) e. B)
74	73	adantr 425	. . . . . . . . . . . . 13 |- (((B e. On /\ A e. B) /\ Lim x) -> (/) e. B)
75		oelim 5214	. . . . . . . . . . . . . 14 |- (((B e. On /\ (x e. _V /\ Lim x)) /\ (/) e. B) -> (B ^o x) = U_y e. x (B ^o y))
76	52, 75	mpanlr1 779	. . . . . . . . . . . . 13 |- (((B e. On /\ Lim x) /\ (/) e. B) -> (B ^o x) = U_y e. x (B ^o y))
77	69, 74, 76	syl11anc 524	. . . . . . . . . . . 12 |- (((B e. On /\ A e. B) /\ Lim x) -> (B ^o x) = U_y e. x (B ^o y))
78	77	adantlr 429	. . . . . . . . . . 11 |- ((((B e. On /\ A e. B) /\ (/) e. A) /\ Lim x) -> (B ^o x) = U_y e. x (B ^o y))
79	78	adantlll 432	. . . . . . . . . 10 |- ((((A e. On /\ (B e. On /\ A e. B)) /\ (/) e. A) /\ Lim x) -> (B ^o x) = U_y e. x (B ^o y))
80	68, 79	sseq12d 2646	. . . . . . . . 9 |- ((((A e. On /\ (B e. On /\ A e. B)) /\ (/) e. A) /\ Lim x) -> ((A ^o x) C_ (B ^o x) <-> U_y e. x (A ^o y) C_ U_y e. x (B ^o y)))
81		ss2iun 3271	. . . . . . . . 9 |- (A.y e. x (A ^o y) C_ (B ^o y) -> U_y e. x (A ^o y) C_ U_y e. x (B ^o y))
82	80, 81	syl5bir 227	. . . . . . . 8 |- ((((A e. On /\ (B e. On /\ A e. B)) /\ (/) e. A) /\ Lim x) -> (A.y e. x (A ^o y) C_ (B ^o y) -> (A ^o x) C_ (B ^o x)))
83	82	ex 402	. . . . . . 7 |- (((A e. On /\ (B e. On /\ A e. B)) /\ (/) e. A) -> (Lim x -> (A.y e. x (A ^o y) C_ (B ^o y) -> (A ^o x) C_ (B ^o x))))
84	64, 83	oe0lem 5197	. . . . . 6 |- ((A e. On /\ (B e. On /\ A e. B)) -> (Lim x -> (A.y e. x (A ^o y) C_ (B ^o y) -> (A ^o x) C_ (B ^o x))))
85	84	com12 14	. . . . 5 |- (Lim x -> ((A e. On /\ (B e. On /\ A e. B)) -> (A.y e. x (A ^o y) C_ (B ^o y) -> (A ^o x) C_ (B ^o x))))
86	13	ancri 321	. . . . 5 |- ((B e. On /\ A e. B) -> (A e. On /\ (B e. On /\ A e. B)))
87	85, 86	syl5 20	. . . 4 |- (Lim x -> ((B e. On /\ A e. B) -> (A.y e. x (A ^o y) C_ (B ^o y) -> (A ^o x) C_ (B ^o x))))
88	3, 6, 9, 12, 20, 49, 87	tfinds3 3948	. . 3 |- (C e. On -> ((B e. On /\ A e. B) -> (A ^o C) C_ (B ^o C)))
89	88	exp3a 405	. 2 |- (C e. On -> (B e. On -> (A e. B -> (A ^o C) C_ (B ^o C))))
90	89	impcom 378	1 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o C) C_ (B ^o C)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292   C_ wss 2593  (/)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:  oeordsuc 5269

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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