Theorem infcdaabs 9028

Description: Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infcdaabs	|- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A +c B ) ~~ A )

Proof of Theorem infcdaabs

Step	Hyp	Ref	Expression
1		cdadom2 9009	. . . . . 6 |- ( B ~<_ A -> ( A +c B ) ~<_ ( A +c A ) )
2	1	3ad2ant3 1084	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A +c B ) ~<_ ( A +c A ) )
3		simp1 1061	. . . . . 6 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> A e. dom card )
4		xp2cda 9002	. . . . . 6 |- ( A e. dom card -> ( A X. 2o ) = ( A +c A ) )
5	3, 4	syl 17	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A X. 2o ) = ( A +c A ) )
6	2, 5	breqtrrd 4681	. . . 4 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A +c B ) ~<_ ( A X. 2o ) )
7		2onn 7720	. . . . . . 7 |- 2o e. om
8		nnsdom 8551	. . . . . . 7 |- ( 2o e. om -> 2o ~< om )
9		sdomdom 7983	. . . . . . 7 |- ( 2o ~< om -> 2o ~<_ om )
10	7, 8, 9	mp2b 10	. . . . . 6 |- 2o ~<_ om
11		simp2 1062	. . . . . 6 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> om ~<_ A )
12		domtr 8009	. . . . . 6 |- ( ( 2o ~<_ om /\ om ~<_ A ) -> 2o ~<_ A )
13	10, 11, 12	sylancr 695	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> 2o ~<_ A )
14		xpdom2g 8056	. . . . 5 |- ( ( A e. dom card /\ 2o ~<_ A ) -> ( A X. 2o ) ~<_ ( A X. A ) )
15	3, 13, 14	syl2anc 693	. . . 4 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A X. 2o ) ~<_ ( A X. A ) )
16		domtr 8009	. . . 4 |- ( ( ( A +c B ) ~<_ ( A X. 2o ) /\ ( A X. 2o ) ~<_ ( A X. A ) ) -> ( A +c B ) ~<_ ( A X. A ) )
17	6, 15, 16	syl2anc 693	. . 3 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A +c B ) ~<_ ( A X. A ) )
18		infxpidm2 8840	. . . 4 |- ( ( A e. dom card /\ om ~<_ A ) -> ( A X. A ) ~~ A )
19	18	3adant3 1081	. . 3 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A X. A ) ~~ A )
20		domentr 8015	. . 3 |- ( ( ( A +c B ) ~<_ ( A X. A ) /\ ( A X. A ) ~~ A ) -> ( A +c B ) ~<_ A )
21	17, 19, 20	syl2anc 693	. 2 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A +c B ) ~<_ A )
22		reldom 7961	. . . . 5 |- Rel ~<_
23	22	brrelexi 5158	. . . 4 |- ( B ~<_ A -> B e. _V )
24	23	3ad2ant3 1084	. . 3 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> B e. _V )
25		cdadom3 9010	. . 3 |- ( ( A e. dom card /\ B e. _V ) -> A ~<_ ( A +c B ) )
26	3, 24, 25	syl2anc 693	. 2 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> A ~<_ ( A +c B ) )
27		sbth 8080	. 2 |- ( ( ( A +c B ) ~<_ A /\ A ~<_ ( A +c B ) ) -> ( A +c B ) ~~ A )
28	21, 26, 27	syl2anc 693	1 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A +c B ) ~~ A )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   X. cxp 5112   dom cdm 5114  (class class class)co 6650   omcom 7065   2oc2o 7554   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   +c ccda 8989

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  ax-inf2 8538

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-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  infunabs  9029  infcda  9030  infdif  9031