Theorem oeordsuc 4279

Description: Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68.

Assertion

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

Proof of Theorem oeordsuc

Step	Hyp	Ref	Expression
1		onelon 3029	. . . 4 |- ((B e. On /\ A e. B) -> A e. On)
2	1	ex 380	. . 3 |- (B e. On -> (A e. B -> A e. On))
3	2	adantr 398	. 2 |- ((B e. On /\ C e. On) -> (A e. B -> A e. On))
4		oewordri 4277	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o C) (_ (B ^o C)))
5	4	3adant1 809	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (A ^o C) (_ (B ^o C)))
6		omwordri 4261	. . . . . . . . . . 11 |- (((A ^o C) e. On /\ (B ^o C) e. On /\ A e. On) -> ((A ^o C) (_ (B ^o C) -> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
7		oecl 4230	. . . . . . . . . . . 12 |- ((A e. On /\ C e. On) -> (A ^o C) e. On)
8	7	3adant2 810	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o C) e. On)
9		oecl 4230	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> (B ^o C) e. On)
10	9	3adant1 809	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o C) e. On)
11		3simp1 800	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> A e. On)
12	6, 8, 10, 11	syl3anc 870	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o C) (_ (B ^o C) -> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
13	5, 12	syld 27	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
14		oesuc 4224	. . . . . . . . . . 11 |- ((A e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
15	14	3adant2 810	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
16	15	sseq1d 2139	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o suc C) (_ ((B ^o C) .o A) <-> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
17	13, 16	sylibrd 211	. . . . . . . 8 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (A ^o suc C) (_ ((B ^o C) .o A)))
18		on0eln0 3081	. . . . . . . . . . . . . 14 |- (B e. On -> ((/) e. B <-> B =/= (/)))
19		ne0i 2337	. . . . . . . . . . . . . 14 |- (A e. B -> B =/= (/))
20	18, 19	syl5bir 217	. . . . . . . . . . . . 13 |- (B e. On -> (A e. B -> (/) e. B))
21	20	adantr 398	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> (A e. B -> (/) e. B))
22		oen0 4271	. . . . . . . . . . . . 13 |- (((B e. On /\ C e. On) /\ (/) e. B) -> (/) e. (B ^o C))
23	22	ex 380	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> ((/) e. B -> (/) e. (B ^o C)))
24	21, 23	syld 27	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> (/) e. (B ^o C)))
25		omordi 4255	. . . . . . . . . . . . . 14 |- (((B e. On /\ (B ^o C) e. On) /\ (/) e. (B ^o C)) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
26		pm3.26 326	. . . . . . . . . . . . . . 15 |- ((B e. On /\ C e. On) -> B e. On)
27	26, 9	jca 295	. . . . . . . . . . . . . 14 |- ((B e. On /\ C e. On) -> (B e. On /\ (B ^o C) e. On))
28	25, 27	sylan 459	. . . . . . . . . . . . 13 |- (((B e. On /\ C e. On) /\ (/) e. (B ^o C)) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
29	28	ex 380	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> ((/) e. (B ^o C) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B))))
30	29	com23 32	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> ((/) e. (B ^o C) -> ((B ^o C) .o A) e. ((B ^o C) .o B))))
31	24, 30	mpdd 46	. . . . . . . . . 10 |- ((B e. On /\ C e. On) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
32	31	3adant1 809	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
33		oesuc 4224	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
34	33	3adant1 809	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
35	34	eleq2d 1588	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> (((B ^o C) .o A) e. (B ^o suc C) <-> ((B ^o C) .o A) e. ((B ^o C) .o B)))
36	32, 35	sylibrd 211	. . . . . . . 8 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((B ^o C) .o A) e. (B ^o suc C)))
37	17, 36	jcad 611	. . . . . . 7 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C))))
38	37	3expa 845	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) -> (A e. B -> ((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C))))
39		ontr2 3061	. . . . . . . . 9 |- (((A ^o suc C) e. On /\ (B ^o suc C) e. On) -> (((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
40		oecl 4230	. . . . . . . . 9 |- ((A e. On /\ suc C e. On) -> (A ^o suc C) e. On)
41		oecl 4230	. . . . . . . . 9 |- ((B e. On /\ suc C e. On) -> (B ^o suc C) e. On)
42	39, 40, 41	syl2an 465	. . . . . . . 8 |- (((A e. On /\ suc C e. On) /\ (B e. On /\ suc C e. On)) -> (((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
43	42	anandirs 524	. . . . . . 7 |- (((A e. On /\ B e. On) /\ suc C e. On) -> (((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
44		sucelon 3125	. . . . . . 7 |- (C e. On <-> suc C e. On)
45	43, 44	sylan2b 463	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) -> (((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
46	38, 45	syld 27	. . . . 5 |- (((A e. On /\ B e. On) /\ C e. On) -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))
47	46	exp31 385	. . . 4 |- (A e. On -> (B e. On -> (C e. On -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))))
48	47	com4l 39	. . 3 |- (B e. On -> (C e. On -> (A e. B -> (A e. On -> (A ^o suc C) e. (B ^o suc C)))))
49	48	imp 357	. 2 |- ((B e. On /\ C e. On) -> (A e. B -> (A e. On -> (A ^o suc C) e. (B ^o suc C))))
50	3, 49	mpdd 46	1 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))

Syntax hints:   -> wi 3   /\ wa 230   /\ w3a 787   = wceq 997   e. wcel 999   =/= wne 1632   (_ wss 2098  (/)c0 2331  Oncon0 3005  suc csuc 3007  (class class class)co 4021   .o comu 4189   ^o coe 4190

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-rab 1699  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-fv 3255  df-rdg 3990  df-opr 4023  df-oprab 4024  df-1o 4191  df-oadd 4193  df-omul 4194  df-oexp 4195

Copyright terms: Public domain