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Theorem isnumbasgrplem2 29598
Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
isnumbasgrplem2  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp )  ->  S  e.  dom  card )

Proof of Theorem isnumbasgrplem2
Dummy variables  a 
b  c  d  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 29594 . . 3  |-  Base  Fn  _V
2 ssv 3474 . . 3  |-  Grp  C_  _V
3 fvelimab 5846 . . 3  |-  ( (
Base  Fn  _V  /\  Grp  C_ 
_V )  ->  (
( S  u.  (har `  S ) )  e.  ( Base " Grp ) 
<->  E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) ) ) )
41, 2, 3mp2an 672 . 2  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp ) 
<->  E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
5 harcl 7877 . . . . . 6  |-  (har `  S )  e.  On
6 onenon 8220 . . . . . 6  |-  ( (har
`  S )  e.  On  ->  (har `  S
)  e.  dom  card )
75, 6ax-mp 5 . . . . 5  |-  (har `  S )  e.  dom  card
8 xpnum 8222 . . . . 5  |-  ( ( (har `  S )  e.  dom  card  /\  (har `  S )  e.  dom  card )  ->  ( (har `  S )  X.  (har `  S ) )  e. 
dom  card )
97, 7, 8mp2an 672 . . . 4  |-  ( (har
`  S )  X.  (har `  S )
)  e.  dom  card
10 ssun1 3617 . . . . . . . 8  |-  S  C_  ( S  u.  (har `  S ) )
11 simpr 461 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
1210, 11syl5sseqr 3503 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  C_  ( Base `  x ) )
13 fvex 5799 . . . . . . . 8  |-  ( Base `  x )  e.  _V
1413ssex 4534 . . . . . . 7  |-  ( S 
C_  ( Base `  x
)  ->  S  e.  _V )
1512, 14syl 16 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  e.  _V )
167a1i 11 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  e.  dom  card )
17 simp1l 1012 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  x  e.  Grp )
18123ad2ant1 1009 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  S  C_  ( Base `  x
) )
19 simp2 989 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  S )
2018, 19sseldd 3455 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  ( Base `  x
) )
21 ssun2 3618 . . . . . . . . . . 11  |-  (har `  S )  C_  ( S  u.  (har `  S
) )
2221, 11syl5sseqr 3503 . . . . . . . . . 10  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  C_  ( Base `  x ) )
23223ad2ant1 1009 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (har `  S )  C_  ( Base `  x ) )
24 simp3 990 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  (har `  S )
)
2523, 24sseldd 3455 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  ( Base `  x
) )
26 eqid 2451 . . . . . . . . 9  |-  ( Base `  x )  =  (
Base `  x )
27 eqid 2451 . . . . . . . . 9  |-  ( +g  `  x )  =  ( +g  `  x )
2826, 27grpcl 15653 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  a  e.  ( Base `  x )  /\  c  e.  ( Base `  x
) )  ->  (
a ( +g  `  x
) c )  e.  ( Base `  x
) )
2917, 20, 25, 28syl3anc 1219 . . . . . . 7  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (
a ( +g  `  x
) c )  e.  ( Base `  x
) )
30 simp1r 1013 . . . . . . 7  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
3129, 30eleqtrd 2541 . . . . . 6  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (
a ( +g  `  x
) c )  e.  ( S  u.  (har `  S ) ) )
32 simplll 757 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  x  e.  Grp )
3322ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  (har `  S
)  C_  ( Base `  x ) )
34 simprl 755 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  c  e.  (har
`  S ) )
3533, 34sseldd 3455 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  c  e.  (
Base `  x )
)
36 simprr 756 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  d  e.  (har
`  S ) )
3733, 36sseldd 3455 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  d  e.  (
Base `  x )
)
3812ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  S  C_  ( Base `  x ) )
39 simplr 754 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  a  e.  S
)
4038, 39sseldd 3455 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  a  e.  (
Base `  x )
)
4126, 27grplcan 15692 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( c  e.  (
Base `  x )  /\  d  e.  ( Base `  x )  /\  a  e.  ( Base `  x ) ) )  ->  ( ( a ( +g  `  x
) c )  =  ( a ( +g  `  x ) d )  <-> 
c  =  d ) )
4232, 35, 37, 40, 41syl13anc 1221 . . . . . 6  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  ( ( a ( +g  `  x
) c )  =  ( a ( +g  `  x ) d )  <-> 
c  =  d ) )
43 simplll 757 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  ->  x  e.  Grp )
4412ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  ->  S  C_  ( Base `  x
) )
45 simprr 756 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
d  e.  S )
4644, 45sseldd 3455 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
d  e.  ( Base `  x ) )
47 simprl 755 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
a  e.  S )
4844, 47sseldd 3455 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
a  e.  ( Base `  x ) )
4922ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
(har `  S )  C_  ( Base `  x
) )
50 simplr 754 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
b  e.  (har `  S ) )
5149, 50sseldd 3455 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
b  e.  ( Base `  x ) )
5226, 27grprcan 15673 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( d  e.  (
Base `  x )  /\  a  e.  ( Base `  x )  /\  b  e.  ( Base `  x ) ) )  ->  ( ( d ( +g  `  x
) b )  =  ( a ( +g  `  x ) b )  <-> 
d  =  a ) )
5343, 46, 48, 51, 52syl13anc 1221 . . . . . 6  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
( ( d ( +g  `  x ) b )  =  ( a ( +g  `  x
) b )  <->  d  =  a ) )
54 harndom 7880 . . . . . . 7  |-  -.  (har `  S )  ~<_  S
5554a1i 11 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  -.  (har `  S )  ~<_  S )
5615, 16, 16, 31, 42, 53, 55unxpwdom3 29586 . . . . 5  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_*  ( (har `  S
)  X.  (har `  S ) ) )
57 wdomnumr 8335 . . . . . 6  |-  ( ( (har `  S )  X.  (har `  S )
)  e.  dom  card  -> 
( S  ~<_*  ( (har `  S
)  X.  (har `  S ) )  <->  S  ~<_  ( (har
`  S )  X.  (har `  S )
) ) )
589, 57ax-mp 5 . . . . 5  |-  ( S  ~<_*  ( (har `  S )  X.  (har `  S )
)  <->  S  ~<_  ( (har `  S )  X.  (har `  S ) ) )
5956, 58sylib 196 . . . 4  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )
60 numdom 8309 . . . 4  |-  ( ( ( (har `  S
)  X.  (har `  S ) )  e. 
dom  card  /\  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )  ->  S  e.  dom  card )
619, 59, 60sylancr 663 . . 3  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  e.  dom  card )
6261rexlimiva 2932 . 2  |-  ( E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) )  ->  S  e.  dom  card )
634, 62sylbi 195 1  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp )  ->  S  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3068    u. cun 3424    C_ wss 3426   class class class wbr 4390   Oncon0 4817    X. cxp 4936   dom cdm 4938   "cima 4941    Fn wfn 5511   ` cfv 5516  (class class class)co 6190    ~<_ cdom 7408  harchar 7872    ~<_* cwdom 7873   cardccrd 8206   Basecbs 14276   +g cplusg 14340   Grpcgrp 15512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-omul 7025  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-oi 7825  df-har 7874  df-wdom 7875  df-card 8210  df-acn 8213  df-slot 14280  df-base 14281  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648
This theorem is referenced by:  isnumbasabl  29600  isnumbasgrp  29601
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