Theorem txopn 15913

Description: The product of two open sets is open in the product topology.

Hypothesis

Ref	Expression
txopn.1	|- T = (R X.t S)

Assertion

Ref	Expression
txopn	|- (((R e. Top /\ S e. Top) /\ (A e. R /\ B e. S)) -> (A X. B) e. T)

Proof of Theorem txopn

Step	Hyp	Ref	Expression
1		txbas 8933	. . . . 5 |- ((R e. Top /\ S e. Top) -> {z | E.u e. R E.v e. S z = (u X. v)} e. Bases)
2		bastg 8892	. . . . 5 |- ({z | E.u e. R E.v e. S z = (u X. v)} e. Bases -> {z | E.u e. R E.v e. S z = (u X. v)} C_ (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
3	1, 2	syl 12	. . . 4 |- ((R e. Top /\ S e. Top) -> {z | E.u e. R E.v e. S z = (u X. v)} C_ (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
4	3	adantr 425	. . 3 |- (((R e. Top /\ S e. Top) /\ (A e. R /\ B e. S)) -> {z | E.u e. R E.v e. S z = (u X. v)} C_ (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
5		eqid 1884	. . . . . 6 |- (A X. B) = (A X. B)
6		xpeq1 4016	. . . . . . . 8 |- (u = A -> (u X. v) = (A X. v))
7	6	eqeq2d 1895	. . . . . . 7 |- (u = A -> ((A X. B) = (u X. v) <-> (A X. B) = (A X. v)))
8		xpeq2 4017	. . . . . . . 8 |- (v = B -> (A X. v) = (A X. B))
9	8	eqeq2d 1895	. . . . . . 7 |- (v = B -> ((A X. B) = (A X. v) <-> (A X. B) = (A X. B)))
10	7, 9	rcla42ev 2385	. . . . . 6 |- ((A e. R /\ B e. S /\ (A X. B) = (A X. B)) -> E.u e. R E.v e. S (A X. B) = (u X. v))
11	5, 10	mp3an3 1180	. . . . 5 |- ((A e. R /\ B e. S) -> E.u e. R E.v e. S (A X. B) = (u X. v))
12		xpexg 4095	. . . . . 6 |- ((A e. R /\ B e. S) -> (A X. B) e. _V)
13		eqeq1 1890	. . . . . . . 8 |- (z = (A X. B) -> (z = (u X. v) <-> (A X. B) = (u X. v)))
14	13	2rexbidv 2141	. . . . . . 7 |- (z = (A X. B) -> (E.u e. R E.v e. S z = (u X. v) <-> E.u e. R E.v e. S (A X. B) = (u X. v)))
15	14	elabg 2405	. . . . . 6 |- ((A X. B) e. _V -> ((A X. B) e. {z | E.u e. R E.v e. S z = (u X. v)} <-> E.u e. R E.v e. S (A X. B) = (u X. v)))
16	12, 15	syl 12	. . . . 5 |- ((A e. R /\ B e. S) -> ((A X. B) e. {z | E.u e. R E.v e. S z = (u X. v)} <-> E.u e. R E.v e. S (A X. B) = (u X. v)))
17	11, 16	mpbird 213	. . . 4 |- ((A e. R /\ B e. S) -> (A X. B) e. {z | E.u e. R E.v e. S z = (u X. v)})
18	17	adantl 424	. . 3 |- (((R e. Top /\ S e. Top) /\ (A e. R /\ B e. S)) -> (A X. B) e. {z | E.u e. R E.v e. S z = (u X. v)})
19	4, 18	sseldd 2620	. 2 |- (((R e. Top /\ S e. Top) /\ (A e. R /\ B e. S)) -> (A X. B) e. (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
20		txval 8932	. . . 4 |- ((R e. Top /\ S e. Top) -> (R X.t S) = (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
21		txopn.1	. . . 4 |- T = (R X.t S)
22	20, 21	syl5eq 1940	. . 3 |- ((R e. Top /\ S e. Top) -> T = (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
23	22	adantr 425	. 2 |- (((R e. Top /\ S e. Top) /\ (A e. R /\ B e. S)) -> T = (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
24	19, 23	eleqtrrd 1974	1 |- (((R e. Top /\ S e. Top) /\ (A e. R /\ B e. S)) -> (A X. B) e. T)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292   C_ wss 2593   X. cxp 3984  ` cfv 3998  (class class class)co 4884  Topctop 8857  Basesctb 8859  topGenctg 8860   X.t ctx 8930

This theorem is referenced by:  txcld 15914

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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-ral 2109  df-rex 2110  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-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-opr 4886  df-oprab 4887  df-top 8861  df-bases 8863  df-topgen 8864  df-tx 8931

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