Theorem txtopon 23620

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 22940	. . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2		topontop 22940	. . 3 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
3		txtop 23598	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top)
5		eqid 2740	. . . . 5 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
6		eqid 2740	. . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅
7		eqid 2740	. . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆
8	5, 6, 7	txuni2 23594	. . . 4 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
9		toponuni 22941	. . . . 5 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
10		toponuni 22941	. . . . 5 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆)
11		xpeq12 5725	. . . . 5 ⊢ ((𝑋 = ∪ 𝑅 ∧ 𝑌 = ∪ 𝑆) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆))
12	9, 10, 11	syl2an 595	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆))
13	5	txbasex 23595	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
14		unitg 22995	. . . . 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 17	. . . 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 2806	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
17	5	txval 23593	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
18	17	unieqd 4944	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (𝑅 ×t 𝑆) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
19	16, 18	eqtr4d 2783	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
20		istopon 22939	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)))
21	4, 19, 20	sylanbrc 582	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∪ cuni 4931   × cxp 5698  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  topGenctg 17497  Topctop 22920  TopOnctopon 22937   ×_t ctx 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974  df-tx 23591

This theorem is referenced by:  txuni  23621  txcls  23633  tx1cn  23638  tx2cn  23639  txcnp  23649  txcnmpt  23653  txindis  23663  txdis1cn  23664  txlm  23677  lmcn2  23678  xkococn  23689  cnmpt12  23696  cnmpt2c  23699  cnmpt21  23700  cnmpt2t  23702  cnmpt22  23703  cnmpt22f  23704  cnmpt2res  23706  cnmptcom  23707  cnmpt2k  23717  ptunhmeo  23837  xpstopnlem1  23838  xkocnv  23843  xkohmeo  23844  txflf  24035  flfcnp2  24036  cnmpt2plusg  24117  tmdcn2  24118  indistgp  24129  clssubg  24138  qustgplem  24150  prdstmdd  24153  tsmsadd  24176  cnmpt2vsca  24224  txmetcn  24582  cnmpt2ds  24884  fsum2cn  24914  cnmpopc  24974  htpyco2  25030  phtpyco2  25041  cnmpt2ip  25301  limccnp2  25947  dvcnp2  25975  dvcnp2OLD  25976  dvaddbr  25994  dvmulbr  25995  dvmulbrOLD  25996  dvcobr  26003  dvcobrOLD  26004  lhop1lem  26072  taylthlem2  26434  taylthlem2OLD  26435  cxpcn3  26809  tpr2tp  33850  txsconnlem  35208  txsconn  35209  cvmlift2lem11  35281  cvmlift2lem12  35282