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Theorem isnumbasgrplem2 31028
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 31024 . . 3  |-  Base  Fn  _V
2 ssv 3509 . . 3  |-  Grp  C_  _V
3 fvelimab 5914 . . 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 7990 . . . . . 6  |-  (har `  S )  e.  On
6 onenon 8333 . . . . . 6  |-  ( (har
`  S )  e.  On  ->  (har `  S
)  e.  dom  card )
75, 6ax-mp 5 . . . . 5  |-  (har `  S )  e.  dom  card
8 xpnum 8335 . . . . 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 3652 . . . . . . . 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 3538 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  C_  ( Base `  x ) )
13 fvex 5866 . . . . . . . 8  |-  ( Base `  x )  e.  _V
1413ssex 4581 . . . . . . 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 1021 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  x  e.  Grp )
18123ad2ant1 1018 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  S  C_  ( Base `  x
) )
19 simp2 998 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  S )
2018, 19sseldd 3490 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  ( Base `  x
) )
21 ssun2 3653 . . . . . . . . . . 11  |-  (har `  S )  C_  ( S  u.  (har `  S
) )
2221, 11syl5sseqr 3538 . . . . . . . . . 10  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  C_  ( Base `  x ) )
23223ad2ant1 1018 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (har `  S )  C_  ( Base `  x ) )
24 simp3 999 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  (har `  S )
)
2523, 24sseldd 3490 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  ( Base `  x
) )
26 eqid 2443 . . . . . . . . 9  |-  ( Base `  x )  =  (
Base `  x )
27 eqid 2443 . . . . . . . . 9  |-  ( +g  `  x )  =  ( +g  `  x )
2826, 27grpcl 15937 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  a  e.  ( Base `  x )  /\  c  e.  ( Base `  x
) )  ->  (
a ( +g  `  x
) c )  e.  ( Base `  x
) )
2917, 20, 25, 28syl3anc 1229 . . . . . . 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 1022 . . . . . . 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 2533 . . . . . 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 759 . . . . . . 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 756 . . . . . . . 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 3490 . . . . . . 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 757 . . . . . . . 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 3490 . . . . . . 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 755 . . . . . . . 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 3490 . . . . . . 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 15976 . . . . . . 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 1231 . . . . . 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 759 . . . . . . 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 757 . . . . . . . 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 3490 . . . . . . 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 756 . . . . . . . 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 3490 . . . . . . 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 755 . . . . . . . 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 3490 . . . . . . 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 15957 . . . . . . 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 1231 . . . . . 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 7993 . . . . . . 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 31016 . . . . 5  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_*  ( (har `  S
)  X.  (har `  S ) ) )
57 wdomnumr 8448 . . . . . 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 8422 . . . 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 2931 . 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 974    = wceq 1383    e. wcel 1804   E.wrex 2794   _Vcvv 3095    u. cun 3459    C_ wss 3461   class class class wbr 4437   Oncon0 4868    X. cxp 4987   dom cdm 4989   "cima 4992    Fn wfn 5573   ` cfv 5578  (class class class)co 6281    ~<_ cdom 7516  harchar 7985    ~<_* cwdom 7986   cardccrd 8319   Basecbs 14509   +g cplusg 14574   Grpcgrp 15927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-omul 7137  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-oi 7938  df-har 7987  df-wdom 7988  df-card 8323  df-acn 8326  df-slot 14513  df-base 14514  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932
This theorem is referenced by:  isnumbasabl  31030  isnumbasgrp  31031
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