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Theorem isnumbasgrplem2 36034
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 36030 . . 3  |-  Base  Fn  _V
2 ssv 3438 . . 3  |-  Grp  C_  _V
3 fvelimab 5936 . . 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 686 . 2  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp ) 
<->  E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
5 harcl 8094 . . . . . 6  |-  (har `  S )  e.  On
6 onenon 8401 . . . . . 6  |-  ( (har
`  S )  e.  On  ->  (har `  S
)  e.  dom  card )
75, 6ax-mp 5 . . . . 5  |-  (har `  S )  e.  dom  card
8 xpnum 8403 . . . . 5  |-  ( ( (har `  S )  e.  dom  card  /\  (har `  S )  e.  dom  card )  ->  ( (har `  S )  X.  (har `  S ) )  e. 
dom  card )
97, 7, 8mp2an 686 . . . 4  |-  ( (har
`  S )  X.  (har `  S )
)  e.  dom  card
10 ssun1 3588 . . . . . . . 8  |-  S  C_  ( S  u.  (har `  S ) )
11 simpr 468 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
1210, 11syl5sseqr 3467 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  C_  ( Base `  x ) )
13 fvex 5889 . . . . . . . 8  |-  ( Base `  x )  e.  _V
1413ssex 4540 . . . . . . 7  |-  ( S 
C_  ( Base `  x
)  ->  S  e.  _V )
1512, 14syl 17 . . . . . 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 1054 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  x  e.  Grp )
18123ad2ant1 1051 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  S  C_  ( Base `  x
) )
19 simp2 1031 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  S )
2018, 19sseldd 3419 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  ( Base `  x
) )
21 ssun2 3589 . . . . . . . . . . 11  |-  (har `  S )  C_  ( S  u.  (har `  S
) )
2221, 11syl5sseqr 3467 . . . . . . . . . 10  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  C_  ( Base `  x ) )
23223ad2ant1 1051 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (har `  S )  C_  ( Base `  x ) )
24 simp3 1032 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  (har `  S )
)
2523, 24sseldd 3419 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  ( Base `  x
) )
26 eqid 2471 . . . . . . . . 9  |-  ( Base `  x )  =  (
Base `  x )
27 eqid 2471 . . . . . . . . 9  |-  ( +g  `  x )  =  ( +g  `  x )
2826, 27grpcl 16757 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  a  e.  ( Base `  x )  /\  c  e.  ( Base `  x
) )  ->  (
a ( +g  `  x
) c )  e.  ( Base `  x
) )
2917, 20, 25, 28syl3anc 1292 . . . . . . 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 1055 . . . . . . 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 2551 . . . . . 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 776 . . . . . . 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 740 . . . . . . . 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 772 . . . . . . . 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 3419 . . . . . . 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 774 . . . . . . . 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 3419 . . . . . . 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 740 . . . . . . . 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 770 . . . . . . . 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 3419 . . . . . . 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 16796 . . . . . . 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 1294 . . . . . 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 776 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  ->  x  e.  Grp )
4412ad2antrr 740 . . . . . . . 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 774 . . . . . . . 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 3419 . . . . . . 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 772 . . . . . . . 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 3419 . . . . . . 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 740 . . . . . . . 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 770 . . . . . . . 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 3419 . . . . . . 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 16777 . . . . . . 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 1294 . . . . . 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 8097 . . . . . . 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 36024 . . . . 5  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_*  ( (har `  S
)  X.  (har `  S ) ) )
57 wdomnumr 8513 . . . . . 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 201 . . . 4  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )
60 numdom 8487 . . . 4  |-  ( ( ( (har `  S
)  X.  (har `  S ) )  e. 
dom  card  /\  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )  ->  S  e.  dom  card )
619, 59, 60sylancr 676 . . 3  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  e.  dom  card )
6261rexlimiva 2868 . 2  |-  ( E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) )  ->  S  e.  dom  card )
634, 62sylbi 200 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    u. cun 3388    C_ wss 3390   class class class wbr 4395    X. cxp 4837   dom cdm 4839   "cima 4842   Oncon0 5430    Fn wfn 5584   ` cfv 5589  (class class class)co 6308    ~<_ cdom 7585  harchar 8089    ~<_* cwdom 8090   cardccrd 8387   Basecbs 15199   +g cplusg 15268   Grpcgrp 16747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-oi 8043  df-har 8091  df-wdom 8092  df-card 8391  df-acn 8394  df-slot 15203  df-base 15204  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752
This theorem is referenced by:  isnumbasabl  36036  isnumbasgrp  36037
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