Theorem isnumbasgrplem2 37674

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.

Step	Hyp	Ref	Expression
1		basfn 15877	. . 3 |- Base Fn _V
2		ssv 3625	. . 3 |- Grp C_ _V
3		fvelimab 6253	. . 3 |- ( ( Base Fn _V /\ Grp C_ _V ) -> ( ( S u. (har ` S ) ) e. ( Base " Grp ) <-> E. x e. Grp ( Base ` x ) = ( S u. (har ` S ) ) ) )
4	1, 2, 3	mp2an 708	. 2 |- ( ( S u. (har ` S ) ) e. ( Base " Grp ) <-> E. x e. Grp ( Base ` x ) = ( S u. (har ` S ) ) )
5		harcl 8466	. . . . . 6 |- (har ` S ) e. On
6		onenon 8775	. . . . . 6 |- ( (har ` S ) e. On -> (har ` S ) e. dom card )
7	5, 6	ax-mp 5	. . . . 5 |- (har ` S ) e. dom card
8		xpnum 8777	. . . . 5 |- ( ( (har ` S ) e. dom card /\ (har ` S ) e. dom card ) -> ( (har ` S ) X. (har ` S ) ) e. dom card )
9	7, 7, 8	mp2an 708	. . . 4 |- ( (har ` S ) X. (har ` S ) ) e. dom card
10		ssun1 3776	. . . . . . . 8 |- S C_ ( S u. (har ` S ) )
11		simpr 477	. . . . . . . 8 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> ( Base ` x ) = ( S u. (har ` S ) ) )
12	10, 11	syl5sseqr 3654	. . . . . . 7 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> S C_ ( Base ` x ) )
13		fvex 6201	. . . . . . . 8 |- ( Base ` x ) e. _V
14	13	ssex 4802	. . . . . . 7 |- ( S C_ ( Base ` x ) -> S e. _V )
15	12, 14	syl 17	. . . . . 6 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> S e. _V )
16	7	a1i 11	. . . . . 6 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> (har ` S ) e. dom card )
17		simp1l 1085	. . . . . . . 8 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> x e. Grp )
18	12	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> S C_ ( Base ` x ) )
19		simp2 1062	. . . . . . . . 9 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> a e. S )
20	18, 19	sseldd 3604	. . . . . . . 8 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> a e. ( Base ` x ) )
21		ssun2 3777	. . . . . . . . . . 11 |- (har ` S ) C_ ( S u. (har ` S ) )
22	21, 11	syl5sseqr 3654	. . . . . . . . . 10 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> (har ` S ) C_ ( Base ` x ) )
23	22	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> (har ` S ) C_ ( Base ` x ) )
24		simp3 1063	. . . . . . . . 9 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> c e. (har ` S ) )
25	23, 24	sseldd 3604	. . . . . . . 8 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> c e. ( Base ` x ) )
26		eqid 2622	. . . . . . . . 9 |- ( Base ` x ) = ( Base ` x )
27		eqid 2622	. . . . . . . . 9 |- ( +g ` x ) = ( +g ` x )
28	26, 27	grpcl 17430	. . . . . . . 8 |- ( ( x e. Grp /\ a e. ( Base ` x ) /\ c e. ( Base ` x ) ) -> ( a ( +g ` x ) c ) e. ( Base ` x ) )
29	17, 20, 25, 28	syl3anc 1326	. . . . . . 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 1086	. . . . . . 7 |- ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S /\ c e. (har ` S ) ) -> ( Base ` x ) = ( S u. (har ` S ) ) )
31	29, 30	eleqtrd 2703	. . . . . 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 798	. . . . . . 7 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S ) /\ ( c e. (har ` S ) /\ d e. (har ` S ) ) ) -> x e. Grp )
33	22	ad2antrr 762	. . . . . . . 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 794	. . . . . . . 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 ) )
35	33, 34	sseldd 3604	. . . . . . 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 796	. . . . . . . 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 ) )
37	33, 36	sseldd 3604	. . . . . . 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 ) )
38	12	ad2antrr 762	. . . . . . . 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 792	. . . . . . . 8 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ a e. S ) /\ ( c e. (har ` S ) /\ d e. (har ` S ) ) ) -> a e. S )
40	38, 39	sseldd 3604	. . . . . . 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 ) )
41	26, 27	grplcan 17477	. . . . . . 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 ) )
42	32, 35, 37, 40, 41	syl13anc 1328	. . . . . 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 798	. . . . . . 7 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ b e. (har ` S ) ) /\ ( a e. S /\ d e. S ) ) -> x e. Grp )
44	12	ad2antrr 762	. . . . . . . 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 796	. . . . . . . 8 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ b e. (har ` S ) ) /\ ( a e. S /\ d e. S ) ) -> d e. S )
46	44, 45	sseldd 3604	. . . . . . 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 794	. . . . . . . 8 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ b e. (har ` S ) ) /\ ( a e. S /\ d e. S ) ) -> a e. S )
48	44, 47	sseldd 3604	. . . . . . 7 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ b e. (har ` S ) ) /\ ( a e. S /\ d e. S ) ) -> a e. ( Base ` x ) )
49	22	ad2antrr 762	. . . . . . . 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 792	. . . . . . . 8 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ b e. (har ` S ) ) /\ ( a e. S /\ d e. S ) ) -> b e. (har ` S ) )
51	49, 50	sseldd 3604	. . . . . . 7 |- ( ( ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) /\ b e. (har ` S ) ) /\ ( a e. S /\ d e. S ) ) -> b e. ( Base ` x ) )
52	26, 27	grprcan 17455	. . . . . . 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 ) )
53	43, 46, 48, 51, 52	syl13anc 1328	. . . . . 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 8469	. . . . . . 7 |- -. (har ` S ) ~<_ S
55	54	a1i 11	. . . . . 6 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> -. (har ` S ) ~<_ S )
56	15, 16, 16, 31, 42, 53, 55	unxpwdom3 37665	. . . . 5 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> S ~<_* ( (har ` S ) X. (har ` S ) ) )
57		wdomnumr 8887	. . . . . 6 |- ( ( (har ` S ) X. (har ` S ) ) e. dom card -> ( S ~<_* ( (har ` S ) X. (har ` S ) ) <-> S ~<_ ( (har ` S ) X. (har ` S ) ) ) )
58	9, 57	ax-mp 5	. . . . 5 |- ( S ~<_* ( (har ` S ) X. (har ` S ) ) <-> S ~<_ ( (har ` S ) X. (har ` S ) ) )
59	56, 58	sylib 208	. . . 4 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> S ~<_ ( (har ` S ) X. (har ` S ) ) )
60		numdom 8861	. . . 4 |- ( ( ( (har ` S ) X. (har ` S ) ) e. dom card /\ S ~<_ ( (har ` S ) X. (har ` S ) ) ) -> S e. dom card )
61	9, 59, 60	sylancr 695	. . 3 |- ( ( x e. Grp /\ ( Base ` x ) = ( S u. (har ` S ) ) ) -> S e. dom card )
62	61	rexlimiva 3028	. 2 |- ( E. x e. Grp ( Base ` x ) = ( S u. (har ` S ) ) -> S e. dom card )
63	4, 62	sylbi 207	1 |- ( ( S u. (har ` S ) ) e. ( Base " Grp ) -> S e. dom card )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   "cima 5117   Oncon0 5723   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   ~<_ cdom 7953  harchar 8461   ~<_^* cwdom 8462   cardccrd 8761   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-wdom 8464  df-card 8765  df-acn 8768  df-slot 15861  df-base 15863  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426

This theorem is referenced by:  isnumbasabl  37676  isnumbasgrp  37677