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Related theorems
Unicode version
Theorem reconn 15451
Description: A subset of the reals is connected iff it has the interval property.
Assertion
Ref Expression
reconn |- (A C_ RR -> ((subSp` <.A, (topGen` ran (,))>.) e. Con <-> A.x e. A A.y e. A (x[,]y) C_ A))
Distinct variable group:   x,y,A
Proof of Theorem reconn
StepHypRef Expression
1 reconnlem1 15446 . . 3 |- ((A C_ RR /\ (subSp` <.A, (topGen` ran (,))>.) e. Con) -> A.x e. A A.y e. A (x[,]y) C_ A)
21ex 402 . 2 |- (A C_ RR -> ((subSp` <.A, (topGen` ran (,))>.) e. Con -> A.x e. A A.y e. A (x[,]y) C_ A))
3 n0 2884 . . . . . . 7 |- ((u i^i A) =/= (/) <-> E.b b e. (u i^i A))
4 n0 2884 . . . . . . . . . 10 |- ((v i^i A) =/= (/) <-> E.c c e. (v i^i A))
5 ssel2 2616 . . . . . . . . . . . . . . . . . . . . 21 |- ((A C_ RR /\ b e. A) -> b e. RR)
6 inss2 2813 . . . . . . . . . . . . . . . . . . . . . 22 |- (u i^i A) C_ A
76sseli 2617 . . . . . . . . . . . . . . . . . . . . 21 |- (b e. (u i^i A) -> b e. A)
85, 7sylan2 500 . . . . . . . . . . . . . . . . . . . 20 |- ((A C_ RR /\ b e. (u i^i A)) -> b e. RR)
983ad2antr1 1041 . . . . . . . . . . . . . . . . . . 19 |- ((A C_ RR /\ (b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A))) -> b e. RR)
10 ssel2 2616 . . . . . . . . . . . . . . . . . . . . 21 |- ((A C_ RR /\ c e. A) -> c e. RR)
11 inss2 2813 . . . . . . . . . . . . . . . . . . . . . 22 |- (v i^i A) C_ A
1211sseli 2617 . . . . . . . . . . . . . . . . . . . . 21 |- (c e. (v i^i A) -> c e. A)
1310, 12sylan2 500 . . . . . . . . . . . . . . . . . . . 20 |- ((A C_ RR /\ c e. (v i^i A)) -> c e. RR)
14133ad2antr2 1042 . . . . . . . . . . . . . . . . . . 19 |- ((A C_ RR /\ (b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A))) -> c e. RR)
15 letric 6802 . . . . . . . . . . . . . . . . . . 19 |- ((b e. RR /\ c e. RR) -> (b <_ c \/ c <_ b))
169, 14, 15syl11anc 524 . . . . . . . . . . . . . . . . . 18 |- ((A C_ RR /\ (b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A))) -> (b <_ c \/ c <_ b))
1716ancoms 484 . . . . . . . . . . . . . . . . 17 |- (((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ A C_ RR) -> (b <_ c \/ c <_ b))
1817adantrr 431 . . . . . . . . . . . . . . . 16 |- (((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) -> (b <_ c \/ c <_ b))
19 opreq1 4889 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (x = b -> (x[,]y) = (b[,]y))
2019sseq1d 2644 . . . . . . . . . . . . . . . . . . . . . . 23 |- (x = b -> ((x[,]y) C_ A <-> (b[,]y) C_ A))
21 opreq2 4890 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = c -> (b[,]y) = (b[,]c))
2221sseq1d 2644 . . . . . . . . . . . . . . . . . . . . . . 23 |- (y = c -> ((b[,]y) C_ A <-> (b[,]c) C_ A))
2320, 22rcla42v 2384 . . . . . . . . . . . . . . . . . . . . . 22 |- ((b e. A /\ c e. A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (b[,]c) C_ A))
2423, 7, 12syl2an 503 . . . . . . . . . . . . . . . . . . . . 21 |- ((b e. (u i^i A) /\ c e. (v i^i A)) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (b[,]c) C_ A))
25243adant3 896 . . . . . . . . . . . . . . . . . . . 20 |- ((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (b[,]c) C_ A))
2625ad2antrr 440 . . . . . . . . . . . . . . . . . . 19 |- ((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (b[,]c) C_ A))
27 biid 187 . . . . . . . . . . . . . . . . . . . . . 22 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) <-> ((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A))
28 eqid 1884 . . . . . . . . . . . . . . . . . . . . . 22 |- sup((u i^i (b[,]c)), RR, < ) = sup((u i^i (b[,]c)), RR, < )
2927, 28reconnlem4 15449 . . . . . . . . . . . . . . . . . . . . 21 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> -. sup((u i^i (b[,]c)), RR, < ) e. u)
3027, 28reconnlem5 15450 . . . . . . . . . . . . . . . . . . . . 21 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> -. sup((u i^i (b[,]c)), RR, < ) e. v)
31 sstr2 2623 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ((b[,]c) C_ A -> (A C_ (u u. v) -> (b[,]c) C_ (u u. v)))
3231adantl 424 . . . . . . . . . . . . . . . . . . . . . . 23 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> (A C_ (u u. v) -> (b[,]c) C_ (u u. v)))
33 ssel 2615 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ((b[,]c) C_ (u u. v) -> (sup((u i^i (b[,]c)), RR, < ) e. (b[,]c) -> sup((u i^i (b[,]c)), RR, < ) e. (u u. v)))
3427, 28reconnlem3 15448 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> sup((u i^i (b[,]c)), RR, < ) e. (b[,]c))
3533, 34syl5com 63 . . . . . . . . . . . . . . . . . . . . . . 23 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> ((b[,]c) C_ (u u. v) -> sup((u i^i (b[,]c)), RR, < ) e. (u u. v)))
3632, 35syld 30 . . . . . . . . . . . . . . . . . . . . . 22 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> (A C_ (u u. v) -> sup((u i^i (b[,]c)), RR, < ) e. (u u. v)))
37 elun 2741 . . . . . . . . . . . . . . . . . . . . . 22 |- (sup((u i^i (b[,]c)), RR, < ) e. (u u. v) <-> (sup((u i^i (b[,]c)), RR, < ) e. u \/ sup((u i^i (b[,]c)), RR, < ) e. v))
3836, 37syl6ib 229 . . . . . . . . . . . . . . . . . . . . 21 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> (A C_ (u u. v) -> (sup((u i^i (b[,]c)), RR, < ) e. u \/ sup((u i^i (b[,]c)), RR, < ) e. v)))
3929, 30, 38mtord 15346 . . . . . . . . . . . . . . . . . . . 20 |- (((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) /\ (b[,]c) C_ A) -> -. A C_ (u u. v))
4039ex 402 . . . . . . . . . . . . . . . . . . 19 |- ((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) -> ((b[,]c) C_ A -> -. A C_ (u u. v)))
4126, 40syld 30 . . . . . . . . . . . . . . . . . 18 |- ((((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) /\ b <_ c) -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v)))
4241ex 402 . . . . . . . . . . . . . . . . 17 |- (((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) -> (b <_ c -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v))))
43 opreq1 4889 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x = c -> (x[,]y) = (c[,]y))
4443sseq1d 2644 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (x = c -> ((x[,]y) C_ A <-> (c[,]y) C_ A))
45 opreq2 4890 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y = b -> (c[,]y) = (c[,]b))
4645sseq1d 2644 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = b -> ((c[,]y) C_ A <-> (c[,]b) C_ A))
4744, 46rcla42v 2384 . . . . . . . . . . . . . . . . . . . . . . 23 |- ((c e. A /\ b e. A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (c[,]b) C_ A))
4847, 12, 7syl2an 503 . . . . . . . . . . . . . . . . . . . . . 22 |- ((c e. (v i^i A) /\ b e. (u i^i A)) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (c[,]b) C_ A))
49483adant3 896 . . . . . . . . . . . . . . . . . . . . 21 |- ((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (c[,]b) C_ A))
5049ad2antrr 440 . . . . . . . . . . . . . . . . . . . 20 |- ((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (c[,]b) C_ A))
51 biid 187 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) <-> ((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A))
52 eqid 1884 . . . . . . . . . . . . . . . . . . . . . . . 24 |- sup((v i^i (c[,]b)), RR, < ) = sup((v i^i (c[,]b)), RR, < )
5351, 52reconnlem4 15449 . . . . . . . . . . . . . . . . . . . . . . 23 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> -. sup((v i^i (c[,]b)), RR, < ) e. v)
5451, 52reconnlem5 15450 . . . . . . . . . . . . . . . . . . . . . . 23 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> -. sup((v i^i (c[,]b)), RR, < ) e. u)
55 sstr2 2623 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((c[,]b) C_ A -> (A C_ (v u. u) -> (c[,]b) C_ (v u. u)))
5655adantl 424 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> (A C_ (v u. u) -> (c[,]b) C_ (v u. u)))
57 ssel 2615 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((c[,]b) C_ (v u. u) -> (sup((v i^i (c[,]b)), RR, < ) e. (c[,]b) -> sup((v i^i (c[,]b)), RR, < ) e. (v u. u)))
5851, 52reconnlem3 15448 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> sup((v i^i (c[,]b)), RR, < ) e. (c[,]b))
5957, 58syl5com 63 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> ((c[,]b) C_ (v u. u) -> sup((v i^i (c[,]b)), RR, < ) e. (v u. u)))
6056, 59syld 30 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> (A C_ (v u. u) -> sup((v i^i (c[,]b)), RR, < ) e. (v u. u)))
61 elun 2741 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (sup((v i^i (c[,]b)), RR, < ) e. (v u. u) <-> (sup((v i^i (c[,]b)), RR, < ) e. v \/ sup((v i^i (c[,]b)), RR, < ) e. u))
6260, 61syl6ib 229 . . . . . . . . . . . . . . . . . . . . . . 23 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> (A C_ (v u. u) -> (sup((v i^i (c[,]b)), RR, < ) e. v \/ sup((v i^i (c[,]b)), RR, < ) e. u)))
6353, 54, 62mtord 15346 . . . . . . . . . . . . . . . . . . . . . 22 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> -. A C_ (v u. u))
64 uncom 2744 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (v u. u) = (u u. v)
6564sseq2i 2642 . . . . . . . . . . . . . . . . . . . . . . 23 |- (A C_ (v u. u) <-> A C_ (u u. v))
6665notbii 204 . . . . . . . . . . . . . . . . . . . . . 22 |- (-. A C_ (v u. u) <-> -. A C_ (u u. v))
6763, 66sylib 215 . . . . . . . . . . . . . . . . . . . . 21 |- (((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) /\ (c[,]b) C_ A) -> -. A C_ (u u. v))
6867ex 402 . . . . . . . . . . . . . . . . . . . 20 |- ((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) -> ((c[,]b) C_ A -> -. A C_ (u u. v)))
6950, 68syld 30 . . . . . . . . . . . . . . . . . . 19 |- ((((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) /\ c <_ b) -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v)))
7069ex 402 . . . . . . . . . . . . . . . . . 18 |- (((c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)) /\ (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))) -> (c <_ b -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v))))
71 incom 2787 . . . . . . . . . . . . . . . . . . . . . 22 |- (u i^i v) = (v i^i u)
7271sseq1i 2641 . . . . . . . . . . . . . . . . . . . . 21 |- ((u i^i v) C_ (RR \ A) <-> (v i^i u) C_ (RR \ A))
7372biimpi 168 . . . . . . . . . . . . . . . . . . . 20 |- ((u i^i v) C_ (RR \ A) -> (v i^i u) C_ (RR \ A))
74733anim3i 1055 . . . . . . . . . . . . . . . . . . 19 |- ((c e. (v i^i A) /\ b e. (u i^i A) /\ (u i^i v) C_ (RR \ A)) -> (c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)))
75743com12 1071 . . . . . . . . . . . . . . . . . 18 |- ((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) -> (c e. (v i^i A) /\ b e. (u i^i A) /\ (v i^i u) C_ (RR \ A)))
76 pm3.22 486 . . . . . . . . . . . . . . . . . . 19 |- ((u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))) -> (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,))))
7776anim2i 362 . . . . . . . . . . . . . . . . . 18 |- ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> (A C_ RR /\ (v e. (topGen` ran (,)) /\ u e. (topGen` ran (,)))))
7870, 75, 77syl2an 503 . . . . . . . . . . . . . . . . 17 |- (((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) -> (c <_ b -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v))))
7942, 78jaod 469 . . . . . . . . . . . . . . . 16 |- (((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) -> ((b <_ c \/ c <_ b) -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v))))
8018, 79mpd 29 . . . . . . . . . . . . . . 15 |- (((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) /\ (A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,))))) -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v)))
8180ex 402 . . . . . . . . . . . . . 14 |- ((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> (A.x e. A A.y e. A (x[,]y) C_ A -> -. A C_ (u u. v))))
8281com23 36 . . . . . . . . . . . . 13 |- ((b e. (u i^i A) /\ c e. (v i^i A) /\ (u i^i v) C_ (RR \ A)) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))
83823exp 1066 . . . . . . . . . . . 12 |- (b e. (u i^i A) -> (c e. (v i^i A) -> ((u i^i v) C_ (RR \ A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))))
8483com12 14 . . . . . . . . . . 11 |- (c e. (v i^i A) -> (b e. (u i^i A) -> ((u i^i v) C_ (RR \ A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))))
858419.23aiv 1674 . . . . . . . . . 10 |- (E.c c e. (v i^i A) -> (b e. (u i^i A) -> ((u i^i v) C_ (RR \ A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))))
864, 85sylbi 216 . . . . . . . . 9 |- ((v i^i A) =/= (/) -> (b e. (u i^i A) -> ((u i^i v) C_ (RR \ A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))))
8786com12 14 . . . . . . . 8 |- (b e. (u i^i A) -> ((v i^i A) =/= (/) -> ((u i^i v) C_ (RR \ A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))))
888719.23aiv 1674 . . . . . . 7 |- (E.b b e. (u i^i A) -> ((v i^i A) =/= (/) -> ((u i^i v) C_ (RR \ A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))))
893, 88sylbi 216 . . . . . 6 |- ((u i^i A) =/= (/) -> ((v i^i A) =/= (/) -> ((u i^i v) C_ (RR \ A) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))))
90893imp 1061 . . . . 5 |- (((u i^i A) =/= (/) /\ (v i^i A) =/= (/) /\ (u i^i v) C_ (RR \ A)) -> (A.x e. A A.y e. A (x[,]y) C_ A -> ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> -. A C_ (u u. v))))
9190com13 37 . . . 4 |- ((A C_ RR /\ (u e. (topGen` ran (,)) /\ v e. (topGen` ran (,)))) -> (A.x e. A A.y e. A (x[,]y) C_ A -> (((u i^i A) =/= (/) /\ (v i^i A) =/= (/) /\ (u i^i v) C_ (RR \ A)) -> -. A C_ (u u. v))))
9291r19.21advva 2185 . . 3 |- (A C_ RR -> (A.x e. A A.y e. A (x[,]y) C_ A -> A.u e. (topGen` ran (,))A.v e. (topGen` ran (,))(((u i^i A) =/= (/) /\ (v i^i A) =/= (/) /\ (u i^i v) C_ (RR \ A)) -> -. A C_ (u u. v))))
93 retop 8926 . . . 4 |- (topGen` ran (,)) e. Top
94 uniretop 8927 . . . . . 6 |- U.(topGen` ran (,)) = RR
9594eqcomi 1888 . . . . 5 |- RR = U.(topGen` ran (,))
9695connsub 15443 . . . 4 |- (((topGen` ran (,)) e. Top /\ A C_ RR) -> ((subSp` <.A, (topGen` ran (,))>.) e. Con <-> A.u e. (topGen` ran (,))A.v e. (topGen` ran (,))(((u i^i A) =/= (/) /\ (v i^i A) =/= (/) /\ (u i^i v) C_ (RR \ A)) -> -. A C_ (u u. v))))
9793, 96mpan 759 . . 3 |- (A C_ RR -> ((subSp` <.A, (topGen` ran (,))>.) e. Con <-> A.u e. (topGen` ran (,))A.v e. (topGen` ran (,))(((u i^i A) =/= (/) /\ (v i^i A) =/= (/) /\ (u i^i v) C_ (RR \ A)) -> -. A C_ (u u. v))))
9892, 97sylibrd 221 . 2 |- (A C_ RR -> (A.x e. A A.y e. A (x[,]y) C_ A -> (subSp` <.A, (topGen` ran (,))>.) e. Con))
992, 98impbid 574 1 |- (A C_ RR -> ((subSp` <.A, (topGen` ran (,))>.) e. Con <-> A.x e. A A.y e. A (x[,]y) C_ A))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  <.cop 3046  U.cuni 3177   class class class wbr 3338  ran crn 3987  ` cfv 3998  (class class class)co 4884  supcsup 5663  RRcr 6385   <_ cle 6448   < clt 6653  (,)cioo 7524  [,]cicc 7527  Topctop 8857  topGenctg 8860  subSpcsubsp 10242  Conccon 10337
This theorem is referenced by:  retopcon 15452  iccconn 15453  ivthALT 15454
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-q 7436  df-ioo 7528  df-icc 7531  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-top 8861  df-topsp 8862  df-bases 8863  df-topgen 8864  df-cld 8939  df-met 9070  df-bl 9072  df-opn 9073  df-subsp 10243  df-con 10338
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