Theorem supmullem2 10994

Description: Lemma for supmul 10995. (Contributed by Mario Carneiro, 5-Jul-2013.)

Hypotheses

Ref	Expression
supmul.1	|- C = { z | E. v e. A E. b e. B z = ( v x. b ) }
supmul.2	|- ( ph <-> ( ( A. x e. A 0 <_ x /\ A. x e. B 0 <_ x ) /\ ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) )

Assertion

Ref	Expression
supmullem2	|- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )

Distinct variable groups:   A, b, v, x, y, w, z   B, b, v, x, y, w, z   x, C, w   ph, b, w, z

Allowed substitution hints:   ph( x, y, v)   C( y, z, v, b)

Proof of Theorem supmullem2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- w e. _V
2		oveq1 6657	. . . . . . . . 9 |- ( v = a -> ( v x. b ) = ( a x. b ) )
3	2	eqeq2d 2632	. . . . . . . 8 |- ( v = a -> ( z = ( v x. b ) <-> z = ( a x. b ) ) )
4	3	rexbidv 3052	. . . . . . 7 |- ( v = a -> ( E. b e. B z = ( v x. b ) <-> E. b e. B z = ( a x. b ) ) )
5	4	cbvrexv 3172	. . . . . 6 |- ( E. v e. A E. b e. B z = ( v x. b ) <-> E. a e. A E. b e. B z = ( a x. b ) )
6		eqeq1 2626	. . . . . . 7 |- ( z = w -> ( z = ( a x. b ) <-> w = ( a x. b ) ) )
7	6	2rexbidv 3057	. . . . . 6 |- ( z = w -> ( E. a e. A E. b e. B z = ( a x. b ) <-> E. a e. A E. b e. B w = ( a x. b ) ) )
8	5, 7	syl5bb 272	. . . . 5 |- ( z = w -> ( E. v e. A E. b e. B z = ( v x. b ) <-> E. a e. A E. b e. B w = ( a x. b ) ) )
9		supmul.1	. . . . 5 |- C = { z | E. v e. A E. b e. B z = ( v x. b ) }
10	1, 8, 9	elab2 3354	. . . 4 |- ( w e. C <-> E. a e. A E. b e. B w = ( a x. b ) )
11		supmul.2	. . . . . . . . . . 11 |- ( ph <-> ( ( A. x e. A 0 <_ x /\ A. x e. B 0 <_ x ) /\ ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) )
12	11	simp2bi 1077	. . . . . . . . . 10 |- ( ph -> ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) )
13	12	simp1d 1073	. . . . . . . . 9 |- ( ph -> A C_ RR )
14	13	sseld 3602	. . . . . . . 8 |- ( ph -> ( a e. A -> a e. RR ) )
15	11	simp3bi 1078	. . . . . . . . . 10 |- ( ph -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
16	15	simp1d 1073	. . . . . . . . 9 |- ( ph -> B C_ RR )
17	16	sseld 3602	. . . . . . . 8 |- ( ph -> ( b e. B -> b e. RR ) )
18	14, 17	anim12d 586	. . . . . . 7 |- ( ph -> ( ( a e. A /\ b e. B ) -> ( a e. RR /\ b e. RR ) ) )
19		remulcl 10021	. . . . . . 7 |- ( ( a e. RR /\ b e. RR ) -> ( a x. b ) e. RR )
20	18, 19	syl6 35	. . . . . 6 |- ( ph -> ( ( a e. A /\ b e. B ) -> ( a x. b ) e. RR ) )
21		eleq1a 2696	. . . . . 6 |- ( ( a x. b ) e. RR -> ( w = ( a x. b ) -> w e. RR ) )
22	20, 21	syl6 35	. . . . 5 |- ( ph -> ( ( a e. A /\ b e. B ) -> ( w = ( a x. b ) -> w e. RR ) ) )
23	22	rexlimdvv 3037	. . . 4 |- ( ph -> ( E. a e. A E. b e. B w = ( a x. b ) -> w e. RR ) )
24	10, 23	syl5bi 232	. . 3 |- ( ph -> ( w e. C -> w e. RR ) )
25	24	ssrdv 3609	. 2 |- ( ph -> C C_ RR )
26	12	simp2d 1074	. . . . 5 |- ( ph -> A =/= (/) )
27	15	simp2d 1074	. . . . . . . 8 |- ( ph -> B =/= (/) )
28		ovex 6678	. . . . . . . . . 10 |- ( a x. b ) e. _V
29	28	isseti 3209	. . . . . . . . 9 |- E. w w = ( a x. b )
30	29	rgenw 2924	. . . . . . . 8 |- A. b e. B E. w w = ( a x. b )
31		r19.2z 4060	. . . . . . . 8 |- ( ( B =/= (/) /\ A. b e. B E. w w = ( a x. b ) ) -> E. b e. B E. w w = ( a x. b ) )
32	27, 30, 31	sylancl 694	. . . . . . 7 |- ( ph -> E. b e. B E. w w = ( a x. b ) )
33		rexcom4 3225	. . . . . . 7 |- ( E. b e. B E. w w = ( a x. b ) <-> E. w E. b e. B w = ( a x. b ) )
34	32, 33	sylib 208	. . . . . 6 |- ( ph -> E. w E. b e. B w = ( a x. b ) )
35	34	ralrimivw 2967	. . . . 5 |- ( ph -> A. a e. A E. w E. b e. B w = ( a x. b ) )
36		r19.2z 4060	. . . . 5 |- ( ( A =/= (/) /\ A. a e. A E. w E. b e. B w = ( a x. b ) ) -> E. a e. A E. w E. b e. B w = ( a x. b ) )
37	26, 35, 36	syl2anc 693	. . . 4 |- ( ph -> E. a e. A E. w E. b e. B w = ( a x. b ) )
38		rexcom4 3225	. . . 4 |- ( E. a e. A E. w E. b e. B w = ( a x. b ) <-> E. w E. a e. A E. b e. B w = ( a x. b ) )
39	37, 38	sylib 208	. . 3 |- ( ph -> E. w E. a e. A E. b e. B w = ( a x. b ) )
40		n0 3931	. . . 4 |- ( C =/= (/) <-> E. w w e. C )
41	10	exbii 1774	. . . 4 |- ( E. w w e. C <-> E. w E. a e. A E. b e. B w = ( a x. b ) )
42	40, 41	bitri 264	. . 3 |- ( C =/= (/) <-> E. w E. a e. A E. b e. B w = ( a x. b ) )
43	39, 42	sylibr 224	. 2 |- ( ph -> C =/= (/) )
44		suprcl 10983	. . . . 5 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR )
45	12, 44	syl 17	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
46		suprcl 10983	. . . . 5 |- ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) -> sup ( B , RR , < ) e. RR )
47	15, 46	syl 17	. . . 4 |- ( ph -> sup ( B , RR , < ) e. RR )
48	45, 47	remulcld 10070	. . 3 |- ( ph -> ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) e. RR )
49	9, 11	supmullem1 10993	. . 3 |- ( ph -> A. w e. C w <_ ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) )
50		breq2 4657	. . . . 5 |- ( x = ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) -> ( w <_ x <-> w <_ ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) ) )
51	50	ralbidv 2986	. . . 4 |- ( x = ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) -> ( A. w e. C w <_ x <-> A. w e. C w <_ ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) ) )
52	51	rspcev 3309	. . 3 |- ( ( ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) e. RR /\ A. w e. C w <_ ( sup ( A , RR , < ) x. sup ( B , RR , < ) ) ) -> E. x e. RR A. w e. C w <_ x )
53	48, 49, 52	syl2anc 693	. 2 |- ( ph -> E. x e. RR A. w e. C w <_ x )
54	25, 43, 53	3jca 1242	1 |- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075

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  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-nel 2898  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  supmul  10995  sqrlem5  13987