Theorem cdalepw 9018

Description: If A is idempotent under cardinal sum and B is dominated by the power set of A, then so is the cardinal sum of A and B. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
cdalepw	|- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ~P A )

Proof of Theorem cdalepw

Step	Hyp	Ref	Expression
1		oveq1 6657	. . 3 |- ( A = (/) -> ( A +c B ) = ( (/) +c B ) )
2	1	breq1d 4663	. 2 |- ( A = (/) -> ( ( A +c B ) ~<_ ~P A <-> ( (/) +c B ) ~<_ ~P A ) )
3		relen 7960	. . . . . . . . 9 |- Rel ~~
4	3	brrelex2i 5159	. . . . . . . 8 |- ( ( A +c A ) ~~ A -> A e. _V )
5	4	adantr 481	. . . . . . 7 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> A e. _V )
6		canth2g 8114	. . . . . . 7 |- ( A e. _V -> A ~< ~P A )
7		sdomdom 7983	. . . . . . 7 |- ( A ~< ~P A -> A ~<_ ~P A )
8	5, 6, 7	3syl 18	. . . . . 6 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> A ~<_ ~P A )
9		simpr 477	. . . . . 6 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> B ~<_ ~P A )
10		cdadom1 9008	. . . . . . 7 |- ( A ~<_ ~P A -> ( A +c B ) ~<_ ( ~P A +c B ) )
11		cdadom2 9009	. . . . . . 7 |- ( B ~<_ ~P A -> ( ~P A +c B ) ~<_ ( ~P A +c ~P A ) )
12		domtr 8009	. . . . . . 7 |- ( ( ( A +c B ) ~<_ ( ~P A +c B ) /\ ( ~P A +c B ) ~<_ ( ~P A +c ~P A ) ) -> ( A +c B ) ~<_ ( ~P A +c ~P A ) )
13	10, 11, 12	syl2an 494	. . . . . 6 |- ( ( A ~<_ ~P A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ( ~P A +c ~P A ) )
14	8, 9, 13	syl2anc 693	. . . . 5 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ( ~P A +c ~P A ) )
15		pwcda1 9016	. . . . . 6 |- ( A e. _V -> ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) )
16	5, 15	syl 17	. . . . 5 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) )
17		domentr 8015	. . . . 5 |- ( ( ( A +c B ) ~<_ ( ~P A +c ~P A ) /\ ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) ) -> ( A +c B ) ~<_ ~P ( A +c 1o ) )
18	14, 16, 17	syl2anc 693	. . . 4 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ~P ( A +c 1o ) )
19	18	adantr 481	. . 3 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> ( A +c B ) ~<_ ~P ( A +c 1o ) )
20		0sdomg 8089	. . . . . . . . 9 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
21	5, 20	syl 17	. . . . . . . 8 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( (/) ~< A <-> A =/= (/) ) )
22	21	biimpar 502	. . . . . . 7 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> (/) ~< A )
23		0sdom1dom 8158	. . . . . . 7 |- ( (/) ~< A <-> 1o ~<_ A )
24	22, 23	sylib 208	. . . . . 6 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> 1o ~<_ A )
25		cdadom2 9009	. . . . . 6 |- ( 1o ~<_ A -> ( A +c 1o ) ~<_ ( A +c A ) )
26	24, 25	syl 17	. . . . 5 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> ( A +c 1o ) ~<_ ( A +c A ) )
27		simpll 790	. . . . 5 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> ( A +c A ) ~~ A )
28		domentr 8015	. . . . 5 |- ( ( ( A +c 1o ) ~<_ ( A +c A ) /\ ( A +c A ) ~~ A ) -> ( A +c 1o ) ~<_ A )
29	26, 27, 28	syl2anc 693	. . . 4 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> ( A +c 1o ) ~<_ A )
30		pwdom 8112	. . . 4 |- ( ( A +c 1o ) ~<_ A -> ~P ( A +c 1o ) ~<_ ~P A )
31	29, 30	syl 17	. . 3 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> ~P ( A +c 1o ) ~<_ ~P A )
32		domtr 8009	. . 3 |- ( ( ( A +c B ) ~<_ ~P ( A +c 1o ) /\ ~P ( A +c 1o ) ~<_ ~P A ) -> ( A +c B ) ~<_ ~P A )
33	19, 31, 32	syl2anc 693	. 2 |- ( ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A =/= (/) ) -> ( A +c B ) ~<_ ~P A )
34		cdacomen 9003	. . 3 |- ( (/) +c B ) ~~ ( B +c (/) )
35		reldom 7961	. . . . . . 7 |- Rel ~<_
36	35	brrelexi 5158	. . . . . 6 |- ( B ~<_ ~P A -> B e. _V )
37	36	adantl 482	. . . . 5 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> B e. _V )
38		cda0en 9001	. . . . 5 |- ( B e. _V -> ( B +c (/) ) ~~ B )
39		domen1 8102	. . . . 5 |- ( ( B +c (/) ) ~~ B -> ( ( B +c (/) ) ~<_ ~P A <-> B ~<_ ~P A ) )
40	37, 38, 39	3syl 18	. . . 4 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( ( B +c (/) ) ~<_ ~P A <-> B ~<_ ~P A ) )
41	9, 40	mpbird 247	. . 3 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( B +c (/) ) ~<_ ~P A )
42		endomtr 8014	. . 3 |- ( ( ( (/) +c B ) ~~ ( B +c (/) ) /\ ( B +c (/) ) ~<_ ~P A ) -> ( (/) +c B ) ~<_ ~P A )
43	34, 41, 42	sylancr 695	. 2 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( (/) +c B ) ~<_ ~P A )
44	2, 33, 43	pm2.61ne 2879	1 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ~P A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653  (class class class)co 6650   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   +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-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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-cda 8990

This theorem is referenced by:  gchdomtri  9451