Theorem txtopon 23910

Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
txtopon	⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))

Proof of Theorem txtopon

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 23231	. . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2		topontop 23231	. . 3 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
3		txtop 23888	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top)
5		eqid 2760	. . . . 5 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
6		eqid 2760	. . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅
7		eqid 2760	. . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆
8	5, 6, 7	txuni2 23884	. . . 4 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
9		toponuni 23232	. . . . 5 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
10		toponuni 23232	. . . . 5 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆)
11		xpeq12 5675	. . . . 5 ⊢ ((𝑋 = ∪ 𝑅 ∧ 𝑌 = ∪ 𝑆) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆))
12	9, 10, 11	syl2an 608	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆))
13	5	txbasex 23885	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
14		unitg 23285	. . . . 5 ⊢ (ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V → ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
15	13, 14	syl 18	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
16	8, 12, 15	3eqtr4a 2821	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
17	5	txval 23883	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
18	17	unieqd 4879	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (𝑅 ×t 𝑆) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
19	16, 18	eqtr4d 2798	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
20		istopon 23230	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)))
21	4, 19, 20	sylanbrc 595	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∪ cuni 4866   × cxp 5648  ran crn 5651  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  topGenctg 17608  Topctop 23211  TopOnctopon 23228   ×_t ctx 23879

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-tx 23881

This theorem is used by:  txuni  23911  txcls  23923  tx1cn  23928  tx2cn  23929  txcnp  23939  txcnmpt  23943  txindis  23953  txdis1cn  23954  txlm  23967  lmcn2  23968  xkococn  23979  cnmpt12  23986  cnmpt2c  23989  cnmpt21  23990  cnmpt2t  23992  cnmpt22  23993  cnmpt22f  23994  cnmpt2res  23996  cnmptcom  23997  cnmpt2k  24007  ptunhmeo  24127  xpstopnlem1  24128  xkocnv  24133  xkohmeo  24134  txflf  24325  flfcnp2  24326  cnmpt2plusg  24407  tmdcn2  24408  indistgp  24419  clssubg  24428  qustgplem  24440  prdstmdd  24443  tsmsadd  24466  cnmpt2vsca  24514  txmetcn  24867  cnmpt2ds  25163  fsum2cn  25192  cnmpopc  25249  htpyco2  25300  phtpyco2  25311  cnmpt2ip  25569  limccnp2  26212  dvcnp2  26240  dvaddbr  26258  dvmulbr  26259  dvcobr  26266  lhop1lem  26333  taylthlem2  26701  cxpcn3  27076  tpr2tp  34536  txsconnlem  36005  txsconn  36006  cvmlift2lem11  36078  cvmlift2lem12  36079