Theorem infxp 9037

Description: Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infxp	|- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( A X. B ) ~~ ( A u. B ) )

Proof of Theorem infxp

Step	Hyp	Ref	Expression
1		sdomdom 7983	. . 3 |- ( B ~< A -> B ~<_ A )
2		infxpabs 9034	. . . . . 6 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B =/= (/) /\ B ~<_ A ) ) -> ( A X. B ) ~~ A )
3		infunabs 9029	. . . . . . . . 9 |- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A u. B ) ~~ A )
4	3	3expa 1265	. . . . . . . 8 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ B ~<_ A ) -> ( A u. B ) ~~ A )
5	4	adantrl 752	. . . . . . 7 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B =/= (/) /\ B ~<_ A ) ) -> ( A u. B ) ~~ A )
6	5	ensymd 8007	. . . . . 6 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B =/= (/) /\ B ~<_ A ) ) -> A ~~ ( A u. B ) )
7		entr 8008	. . . . . 6 |- ( ( ( A X. B ) ~~ A /\ A ~~ ( A u. B ) ) -> ( A X. B ) ~~ ( A u. B ) )
8	2, 6, 7	syl2anc 693	. . . . 5 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B =/= (/) /\ B ~<_ A ) ) -> ( A X. B ) ~~ ( A u. B ) )
9	8	expr 643	. . . 4 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ B =/= (/) ) -> ( B ~<_ A -> ( A X. B ) ~~ ( A u. B ) ) )
10	9	adantrl 752	. . 3 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( B ~<_ A -> ( A X. B ) ~~ ( A u. B ) ) )
11	1, 10	syl5 34	. 2 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( B ~< A -> ( A X. B ) ~~ ( A u. B ) ) )
12		domtri2 8815	. . . 4 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~<_ B <-> -. B ~< A ) )
13	12	ad2ant2r 783	. . 3 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( A ~<_ B <-> -. B ~< A ) )
14		xpcomeng 8052	. . . . . . 7 |- ( ( A e. dom card /\ B e. dom card ) -> ( A X. B ) ~~ ( B X. A ) )
15	14	ad2ant2r 783	. . . . . 6 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( A X. B ) ~~ ( B X. A ) )
16	15	adantr 481	. . . . 5 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> ( A X. B ) ~~ ( B X. A ) )
17		simplrl 800	. . . . . . 7 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> B e. dom card )
18		simplr 792	. . . . . . . 8 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> om ~<_ A )
19		domtr 8009	. . . . . . . 8 |- ( ( om ~<_ A /\ A ~<_ B ) -> om ~<_ B )
20	18, 19	sylan 488	. . . . . . 7 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> om ~<_ B )
21		infn0 8222	. . . . . . . 8 |- ( om ~<_ A -> A =/= (/) )
22	21	ad3antlr 767	. . . . . . 7 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> A =/= (/) )
23		simpr 477	. . . . . . 7 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> A ~<_ B )
24		infxpabs 9034	. . . . . . 7 |- ( ( ( B e. dom card /\ om ~<_ B ) /\ ( A =/= (/) /\ A ~<_ B ) ) -> ( B X. A ) ~~ B )
25	17, 20, 22, 23, 24	syl22anc 1327	. . . . . 6 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> ( B X. A ) ~~ B )
26		uncom 3757	. . . . . . . 8 |- ( A u. B ) = ( B u. A )
27		infunabs 9029	. . . . . . . . 9 |- ( ( B e. dom card /\ om ~<_ B /\ A ~<_ B ) -> ( B u. A ) ~~ B )
28	17, 20, 23, 27	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> ( B u. A ) ~~ B )
29	26, 28	syl5eqbr 4688	. . . . . . 7 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> ( A u. B ) ~~ B )
30	29	ensymd 8007	. . . . . 6 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> B ~~ ( A u. B ) )
31		entr 8008	. . . . . 6 |- ( ( ( B X. A ) ~~ B /\ B ~~ ( A u. B ) ) -> ( B X. A ) ~~ ( A u. B ) )
32	25, 30, 31	syl2anc 693	. . . . 5 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> ( B X. A ) ~~ ( A u. B ) )
33		entr 8008	. . . . 5 |- ( ( ( A X. B ) ~~ ( B X. A ) /\ ( B X. A ) ~~ ( A u. B ) ) -> ( A X. B ) ~~ ( A u. B ) )
34	16, 32, 33	syl2anc 693	. . . 4 |- ( ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) /\ A ~<_ B ) -> ( A X. B ) ~~ ( A u. B ) )
35	34	ex 450	. . 3 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( A ~<_ B -> ( A X. B ) ~~ ( A u. B ) ) )
36	13, 35	sylbird 250	. 2 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( -. B ~< A -> ( A X. B ) ~~ ( A u. B ) ) )
37	11, 36	pm2.61d 170	1 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( A X. B ) ~~ ( A u. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   u. cun 3572   (/)c0 3915   class class class wbr 4653   X. cxp 5112   dom cdm 5114   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761

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:  alephmul  9400