Theorem oewordi 4218

Description: Weak ordering property of ordinal exponentiation.

Assertion

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

Proof of Theorem oewordi

Step	Hyp	Ref	Expression
1		eloni 2958	. . . . . 6 |- (C e. On -> Ord C)
2		ordgt0ge1 4144	. . . . . 6 |- (Ord C -> ((/) e. C <-> 1o (_ C))
3	1, 2	syl 10	. . . . 5 |- (C e. On -> ((/) e. C <-> 1o (_ C))
4		1on 4138	. . . . . 6 |- 1o e. On
5		onsseleq 2999	. . . . . 6 |- ((1o e. On /\ C e. On) -> (1o (_ C <-> (1o e. C \/ 1o = C)))
6	4, 5	mpan 695	. . . . 5 |- (C e. On -> (1o (_ C <-> (1o e. C \/ 1o = C)))
7	3, 6	bitrd 528	. . . 4 |- (C e. On -> ((/) e. C <-> (1o e. C \/ 1o = C)))
8	7	3ad2ant3 802	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> ((/) e. C <-> (1o e. C \/ 1o = C)))
9		oeword 4217	. . . . . 6 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A (_ B <-> (C ^o A) (_ (C ^o B)))
10	9	biimpd 153	. . . . 5 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A (_ B -> (C ^o A) (_ (C ^o B)))
11	10	ex 373	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (1o e. C -> (A (_ B -> (C ^o A) (_ (C ^o B))))
12		opreq1 3968	. . . . . . . 8 |- (1o = C -> (1o ^o A) = (C ^o A))
13		opreq1 3968	. . . . . . . 8 |- (1o = C -> (1o ^o B) = (C ^o B))
14	12, 13	sseq12d 2090	. . . . . . 7 |- (1o = C -> ((1o ^o A) (_ (1o ^o B) <-> (C ^o A) (_ (C ^o B)))
15		oe1m 4179	. . . . . . . . . 10 |- (A e. On -> (1o ^o A) = 1o)
16	15	adantr 389	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (1o ^o A) = 1o)
17		oe1m 4179	. . . . . . . . . 10 |- (B e. On -> (1o ^o B) = 1o)
18	17	adantl 388	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (1o ^o B) = 1o)
19	16, 18	eqtr4d 1510	. . . . . . . 8 |- ((A e. On /\ B e. On) -> (1o ^o A) = (1o ^o B))
20		eqimss 2109	. . . . . . . 8 |- ((1o ^o A) = (1o ^o B) -> (1o ^o A) (_ (1o ^o B))
21	19, 20	syl 10	. . . . . . 7 |- ((A e. On /\ B e. On) -> (1o ^o A) (_ (1o ^o B))
22	14, 21	syl5cbi 209	. . . . . 6 |- ((A e. On /\ B e. On) -> (1o = C -> (C ^o A) (_ (C ^o B)))
23	22	3adant3 799	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) -> (1o = C -> (C ^o A) (_ (C ^o B)))
24	23	a1dd 42	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (1o = C -> (A (_ B -> (C ^o A) (_ (C ^o B))))
25	11, 24	jaod 424	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> ((1o e. C \/ 1o = C) -> (A (_ B -> (C ^o A) (_ (C ^o B))))
26	8, 25	sylbid 203	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> ((/) e. C -> (A (_ B -> (C ^o A) (_ (C ^o B))))
27	26	imp 350	1 |- (((A e. On /\ B e. On /\ C e. On) /\ (/) e. C) -> (A (_ B -> (C ^o A) (_ (C ^o B)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   (_ wss 2047  (/)c0 2280  Ord word 2947  Oncon0 2948  (class class class)co 3963  1oc1o 4128   ^o coe 4132

This theorem is referenced by:  oelim2 4222

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1o 4133  df-oadd 4135  df-omul 4136  df-oexp 4137

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