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Theorem oaabs 5309
Description: Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59.
Assertion
Ref Expression
oaabs |- (((A e. om /\ B e. On) /\ om C_ B) -> (A +o B) = B)

Proof of Theorem oaabs
StepHypRef Expression
1 ssexg 3457 . . . . . . . 8 |- ((om C_ B /\ B e. On) -> om e. _V)
21ex 402 . . . . . . 7 |- (om C_ B -> (B e. On -> om e. _V))
3 ordom 3960 . . . . . . . 8 |- Ord om
4 elong 3665 . . . . . . . 8 |- (om e. _V -> (om e. On <-> Ord om))
53, 4mpbiri 211 . . . . . . 7 |- (om e. _V -> om e. On)
62, 5syl6com 64 . . . . . 6 |- (B e. On -> (om C_ B -> om e. On))
76imp 377 . . . . 5 |- ((B e. On /\ om C_ B) -> om e. On)
8 opreq2 4890 . . . . . . . . 9 |- (x = om -> (A +o x) = (A +o om))
9 id 73 . . . . . . . . 9 |- (x = om -> x = om)
108, 9eqeq12d 1899 . . . . . . . 8 |- (x = om -> ((A +o x) = x <-> (A +o om) = om))
1110imbi2d 674 . . . . . . 7 |- (x = om -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o om) = om)))
12 opreq2 4890 . . . . . . . . 9 |- (x = y -> (A +o x) = (A +o y))
13 id 73 . . . . . . . . 9 |- (x = y -> x = y)
1412, 13eqeq12d 1899 . . . . . . . 8 |- (x = y -> ((A +o x) = x <-> (A +o y) = y))
1514imbi2d 674 . . . . . . 7 |- (x = y -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o y) = y)))
16 opreq2 4890 . . . . . . . . 9 |- (x = suc y -> (A +o x) = (A +o suc y))
17 id 73 . . . . . . . . 9 |- (x = suc y -> x = suc y)
1816, 17eqeq12d 1899 . . . . . . . 8 |- (x = suc y -> ((A +o x) = x <-> (A +o suc y) = suc y))
1918imbi2d 674 . . . . . . 7 |- (x = suc y -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o suc y) = suc y)))
20 opreq2 4890 . . . . . . . . 9 |- (x = B -> (A +o x) = (A +o B))
21 id 73 . . . . . . . . 9 |- (x = B -> x = B)
2220, 21eqeq12d 1899 . . . . . . . 8 |- (x = B -> ((A +o x) = x <-> (A +o B) = B))
2322imbi2d 674 . . . . . . 7 |- (x = B -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o B) = B)))
24 oaabslem 5308 . . . . . . . 8 |- ((om e. On /\ A e. om) -> (A +o om) = om)
2524ex 402 . . . . . . 7 |- (om e. On -> (A e. om -> (A +o om) = om))
26 oasuc 5208 . . . . . . . . . . . . 13 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
27 nnon 3957 . . . . . . . . . . . . 13 |- (A e. om -> A e. On)
2826, 27sylan 497 . . . . . . . . . . . 12 |- ((A e. om /\ y e. On) -> (A +o suc y) = suc (A +o y))
29 suceq 3729 . . . . . . . . . . . 12 |- ((A +o y) = y -> suc (A +o y) = suc y)
3028, 29sylan9eq 1948 . . . . . . . . . . 11 |- (((A e. om /\ y e. On) /\ (A +o y) = y) -> (A +o suc y) = suc y)
3130exp31 407 . . . . . . . . . 10 |- (A e. om -> (y e. On -> ((A +o y) = y -> (A +o suc y) = suc y)))
3231com12 14 . . . . . . . . 9 |- (y e. On -> (A e. om -> ((A +o y) = y -> (A +o suc y) = suc y)))
3332ad2antrr 440 . . . . . . . 8 |- (((y e. On /\ om e. On) /\ om C_ y) -> (A e. om -> ((A +o y) = y -> (A +o suc y) = suc y)))
3433a2d 16 . . . . . . 7 |- (((y e. On /\ om e. On) /\ om C_ y) -> ((A e. om -> (A +o y) = y) -> (A e. om -> (A +o suc y) = suc y)))
35 oalim 5212 . . . . . . . . . . . . . . . . . . 19 |- ((A e. On /\ (om e. On /\ Lim om)) -> (A +o om) = U_y e. om (A +o y))
36 limom 3967 . . . . . . . . . . . . . . . . . . . 20 |- Lim om
3736jctr 315 . . . . . . . . . . . . . . . . . . 19 |- (om e. On -> (om e. On /\ Lim om))
3835, 27, 37syl2an 503 . . . . . . . . . . . . . . . . . 18 |- ((A e. om /\ om e. On) -> (A +o om) = U_y e. om (A +o y))
3924ancoms 484 . . . . . . . . . . . . . . . . . . 19 |- ((A e. om /\ om e. On) -> (A +o om) = om)
40 limuni 3724 . . . . . . . . . . . . . . . . . . . 20 |- (Lim om -> om = U.om)
4136, 40ax-mp 7 . . . . . . . . . . . . . . . . . . 19 |- om = U.om
4239, 41syl6eq 1944 . . . . . . . . . . . . . . . . . 18 |- ((A e. om /\ om e. On) -> (A +o om) = U.om)
4338, 42eqtr3d 1927 . . . . . . . . . . . . . . . . 17 |- ((A e. om /\ om e. On) -> U_y e. om (A +o y) = U.om)
4443adantl 424 . . . . . . . . . . . . . . . 16 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> U_y e. om (A +o y) = U.om)
45 ordelon 3682 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((Ord x /\ y e. x) -> y e. On)
46 limord 3723 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (Lim x -> Ord x)
4745, 46sylan 497 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((Lim x /\ y e. x) -> y e. On)
48 eloni 3667 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y e. On -> Ord y)
49 ordtri1 3693 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((Ord om /\ Ord y) -> (om C_ y <-> -. y e. om))
503, 49mpan 759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (Ord y -> (om C_ y <-> -. y e. om))
5147, 48, 503syl 24 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((Lim x /\ y e. x) -> (om C_ y <-> -. y e. om))
5251pm5.32da 711 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (Lim x -> ((y e. x /\ om C_ y) <-> (y e. x /\ -. y e. om)))
53 eldif 2609 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. (x \ om) <-> (y e. x /\ -. y e. om))
5452, 53syl6bbr 597 . . . . . . . . . . . . . . . . . . . . . . 23 |- (Lim x -> ((y e. x /\ om C_ y) <-> y e. (x \ om)))
5554imbi1d 675 . . . . . . . . . . . . . . . . . . . . . 22 |- (Lim x -> (((y e. x /\ om C_ y) -> (A +o y) = y) <-> (y e. (x \ om) -> (A +o y) = y)))
56 impexp 374 . . . . . . . . . . . . . . . . . . . . . 22 |- (((y e. x /\ om C_ y) -> (A +o y) = y) <-> (y e. x -> (om C_ y -> (A +o y) = y)))
5755, 56syl5bbr 593 . . . . . . . . . . . . . . . . . . . . 21 |- (Lim x -> ((y e. x -> (om C_ y -> (A +o y) = y)) <-> (y e. (x \ om) -> (A +o y) = y)))
5857ralbidv2 2125 . . . . . . . . . . . . . . . . . . . 20 |- (Lim x -> (A.y e. x (om C_ y -> (A +o y) = y) <-> A.y e. (x \ om)(A +o y) = y))
59 iuneq2 3273 . . . . . . . . . . . . . . . . . . . . 21 |- (A.y e. (x \ om)(A +o y) = y -> U_y e. (x \ om)(A +o y) = U_y e. (x \ om)y)
60 uniiun 3306 . . . . . . . . . . . . . . . . . . . . 21 |- U.(x \ om) = U_y e. (x \ om)y
6159, 60syl6eqr 1946 . . . . . . . . . . . . . . . . . . . 20 |- (A.y e. (x \ om)(A +o y) = y -> U_y e. (x \ om)(A +o y) = U.(x \ om))
6258, 61syl6bi 231 . . . . . . . . . . . . . . . . . . 19 |- (Lim x -> (A.y e. x (om C_ y -> (A +o y) = y) -> U_y e. (x \ om)(A +o y) = U.(x \ om)))
6362imp 377 . . . . . . . . . . . . . . . . . 18 |- ((Lim x /\ A.y e. x (om C_ y -> (A +o y) = y)) -> U_y e. (x \ om)(A +o y) = U.(x \ om))
6463adantlr 429 . . . . . . . . . . . . . . . . 17 |- (((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) -> U_y e. (x \ om)(A +o y) = U.(x \ om))
6564adantr 425 . . . . . . . . . . . . . . . 16 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> U_y e. (x \ om)(A +o y) = U.(x \ om))
6644, 65uneq12d 2756 . . . . . . . . . . . . . . 15 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)) = (U.om u. U.(x \ om)))
67 oalim 5212 . . . . . . . . . . . . . . . . . . . 20 |- ((A e. On /\ (x e. _V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
68 visset 2295 . . . . . . . . . . . . . . . . . . . . 21 |- x e. _V
6968jctl 314 . . . . . . . . . . . . . . . . . . . 20 |- (Lim x -> (x e. _V /\ Lim x))
7067, 27, 69syl2an 503 . . . . . . . . . . . . . . . . . . 19 |- ((A e. om /\ Lim x) -> (A +o x) = U_y e. x (A +o y))
7170ancoms 484 . . . . . . . . . . . . . . . . . 18 |- ((Lim x /\ A e. om) -> (A +o x) = U_y e. x (A +o y))
72 ssequn1 2775 . . . . . . . . . . . . . . . . . . . . . 22 |- (om C_ x <-> (om u. x) = x)
7372biimpi 168 . . . . . . . . . . . . . . . . . . . . 21 |- (om C_ x -> (om u. x) = x)
74 undif2 2950 . . . . . . . . . . . . . . . . . . . . 21 |- (om u. (x \ om)) = (om u. x)
7573, 74syl5eq 1940 . . . . . . . . . . . . . . . . . . . 20 |- (om C_ x -> (om u. (x \ om)) = x)
76 iuneq1 3269 . . . . . . . . . . . . . . . . . . . 20 |- ((om u. (x \ om)) = x -> U_y e. (om u. (x \ om))(A +o y) = U_y e. x (A +o y))
7775, 76syl 12 . . . . . . . . . . . . . . . . . . 19 |- (om C_ x -> U_y e. (om u. (x \ om))(A +o y) = U_y e. x (A +o y))
78 iunxun 3329 . . . . . . . . . . . . . . . . . . 19 |- U_y e. (om u. (x \ om))(A +o y) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y))
7977, 78syl5reqr 1943 . . . . . . . . . . . . . . . . . 18 |- (om C_ x -> U_y e. x (A +o y) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
8071, 79sylan9eq 1948 . . . . . . . . . . . . . . . . 17 |- (((Lim x /\ A e. om) /\ om C_ x) -> (A +o x) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
8180an1rs 547 . . . . . . . . . . . . . . . 16 |- (((Lim x /\ om C_ x) /\ A e. om) -> (A +o x) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
8281ad2ant2r 445 . . . . . . . . . . . . . . 15 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> (A +o x) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
83 limuni 3724 . . . . . . . . . . . . . . . . 17 |- (Lim x -> x = U.x)
8475unieqd 3188 . . . . . . . . . . . . . . . . . 18 |- (om C_ x -> U.(om u. (x \ om)) = U.x)
85 uniun 3196 . . . . . . . . . . . . . . . . . 18 |- U.(om u. (x \ om)) = (U.om u. U.(x \ om))
8684, 85syl5reqr 1943 . . . . . . . . . . . . . . . . 17 |- (om C_ x -> U.x = (U.om u. U.(x \ om)))
8783, 86sylan9eq 1948 . . . . . . . . . . . . . . . 16 |- ((Lim x /\ om C_ x) -> x = (U.om u. U.(x \ om)))
8887ad2antrr 440 . . . . . . . . . . . . . . 15 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> x = (U.om u. U.(x \ om)))
8966, 82, 883eqtr4d 1937 . . . . . . . . . . . . . 14 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> (A +o x) = x)
9089exp43 415 . . . . . . . . . . . . 13 |- ((Lim x /\ om C_ x) -> (A.y e. x (om C_ y -> (A +o y) = y) -> (A e. om -> (om e. On -> (A +o x) = x))))
9190com23 36 . . . . . . . . . . . 12 |- ((Lim x /\ om C_ x) -> (A e. om -> (A.y e. x (om C_ y -> (A +o y) = y) -> (om e. On -> (A +o x) = x))))
9291a2d 16 . . . . . . . . . . 11 |- ((Lim x /\ om C_ x) -> ((A e. om -> A.y e. x (om C_ y -> (A +o y) = y)) -> (A e. om -> (om e. On -> (A +o x) = x))))
93 bi2.04 177 . . . . . . . . . . . . 13 |- ((om C_ y -> (A e. om -> (A +o y) = y)) <-> (A e. om -> (om C_ y -> (A +o y) = y)))
9493ralbii 2127 . . . . . . . . . . . 12 |- (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) <-> A.y e. x (A e. om -> (om C_ y -> (A +o y) = y)))
95 r19.21v 2178 . . . . . . . . . . . 12 |- (A.y e. x (A e. om -> (om C_ y -> (A +o y) = y)) <-> (A e. om -> A.y e. x (om C_ y -> (A +o y) = y)))
9694, 95bitri 190 . . . . . . . . . . 11 |- (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) <-> (A e. om -> A.y e. x (om C_ y -> (A +o y) = y)))
9792, 96syl5ib 223 . . . . . . . . . 10 |- ((Lim x /\ om C_ x) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (om e. On -> (A +o x) = x))))
9897com4r 45 . . . . . . . . 9 |- (om e. On -> ((Lim x /\ om C_ x) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (A +o x) = x))))
9998impcom 378 . . . . . . . 8 |- (((Lim x /\ om C_ x) /\ om e. On) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (A +o x) = x)))
10099an1rs 547 . . . . . . 7 |- (((Lim x /\ om e. On) /\ om C_ x) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (A +o x) = x)))
10111, 15, 19, 23, 25, 34, 100tfindsg 3944 . . . . . 6 |- (((B e. On /\ om e. On) /\ om C_ B) -> (A e. om -> (A +o B) = B))
102101an1rs 547 . . . . 5 |- (((B e. On /\ om C_ B) /\ om e. On) -> (A e. om -> (A +o B) = B))
1037, 102mpdan 768 . . . 4 |- ((B e. On /\ om C_ B) -> (A e. om -> (A +o B) = B))
104103ex 402 . . 3 |- (B e. On -> (om C_ B -> (A e. om -> (A +o B) = B)))
105104com3r 39 . 2 |- (A e. om -> (B e. On -> (om C_ B -> (A +o B) = B)))
106105imp31 389 1 |- (((A e. om /\ B e. On) /\ om C_ B) -> (A +o B) = B)
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  U.cuni 3177  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  omcom 3949  (class class class)co 4884   +o coa 5174
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-om 3950  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-oadd 5179
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