Theorem tmdcn2 23958

Description: Write out the definition of continuity of +_g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
tmdcn2.1	⊢ 𝐵 = (Base‘𝐺)
tmdcn2.2	⊢ 𝐽 = (TopOpen‘𝐺)
tmdcn2.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
tmdcn2	⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))

Distinct variable groups:   𝑣,𝑢,𝑥,𝑦,𝐺   𝑢,𝐽,𝑣   𝑢,𝑈,𝑣,𝑥,𝑦   𝑢,𝑋,𝑣   𝑢,𝑌,𝑣

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢)   + (𝑥,𝑦,𝑣,𝑢)   𝐽(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem tmdcn2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tmdcn2.2	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝐺)
2		tmdcn2.1	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	tmdtopon 23950	. . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4	3	ad2antrr 726	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5		eqid 2729	. . . . . 6 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
6	1, 5	tmdcn 23952	. . . . 5 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
7	6	ad2antrr 726	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8		simpr1 1195	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋 ∈ 𝐵)
9		simpr2 1196	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌 ∈ 𝐵)
10	8, 9	opelxpd 5652	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		txtopon 23460	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
12	4, 4, 11	syl2anc 584	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
13		toponuni 22783	. . . . . 6 ⊢ ((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) → (𝐵 × 𝐵) = ∪ (𝐽 ×t 𝐽))
14	12, 13	syl 17	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐵 × 𝐵) = ∪ (𝐽 ×t 𝐽))
15	10, 14	eleqtrd 2830	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ ∪ (𝐽 ×t 𝐽))
16		eqid 2729	. . . . 5 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
17	16	cncnpi 23147	. . . 4 ⊢ (((+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ ⟨𝑋, 𝑌⟩ ∈ ∪ (𝐽 ×t 𝐽)) → (+𝑓‘𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
18	7, 15, 17	syl2anc 584	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓‘𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
19		simplr 768	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑈 ∈ 𝐽)
20		tmdcn2.3	. . . . . 6 ⊢ + = (+g‘𝐺)
21	2, 20, 5	plusfval 18508	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+𝑓‘𝐺)𝑌) = (𝑋 + 𝑌))
22	8, 9, 21	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓‘𝐺)𝑌) = (𝑋 + 𝑌))
23		simpr3 1197	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
24	22, 23	eqeltrd 2828	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓‘𝐺)𝑌) ∈ 𝑈)
25	4, 4, 18, 19, 8, 9, 24	txcnpi 23477	. 2 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈)))
26		dfss3 3920	. . . . . . 7 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈))
27		eleq1 2816	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡(+𝑓‘𝐺) “ 𝑈)))
28	2, 5	plusffn 18510	. . . . . . . . . 10 ⊢ (+𝑓‘𝐺) Fn (𝐵 × 𝐵)
29		elpreima 6985	. . . . . . . . . 10 ⊢ ((+𝑓‘𝐺) Fn (𝐵 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
30	28, 29	ax-mp 5	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
31	27, 30	bitrdi 287	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
32	31	ralxp 5778	. . . . . . 7 ⊢ (∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
33	26, 32	bitri 275	. . . . . 6 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
34		opelxp 5649	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
35		df-ov 7343	. . . . . . . . . . 11 ⊢ (𝑥(+𝑓‘𝐺)𝑦) = ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩)
36	2, 20, 5	plusfval 18508	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+𝑓‘𝐺)𝑦) = (𝑥 + 𝑦))
37	35, 36	eqtr3id 2778	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
38	34, 37	sylbi 217	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
39	38	eleq1d 2813	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
40	39	biimpa 476	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
41	40	2ralimi 3099	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈)
42	33, 41	sylbi 217	. . . . 5 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) → ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈)
43	42	3anim3i 1154	. . . 4 ⊢ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈)) → (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))
44	43	reximi 3067	. . 3 ⊢ (∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈)) → ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))
45	44	reximi 3067	. 2 ⊢ (∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))
46	25, 45	syl 17	1 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3899  ⟨cop 4579  ∪ cuni 4856   × cxp 5611  ^◡ccnv 5612   “ cima 5616   Fn wfn 6471  ‘cfv 6476  (class class class)co 7340  Basecbs 17107  +_gcplusg 17148  TopOpenctopn 17312  +_𝑓cplusf 18498  TopOnctopon 22779   Cn ccn 23093   CnP ccnp 23094   ×_t ctx 23429  TopMndctmd 23939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7343  df-oprab 7344  df-mpo 7345  df-1st 7915  df-2nd 7916  df-map 8746  df-topgen 17334  df-plusf 18500  df-top 22763  df-topon 22780  df-topsp 22802  df-bases 22815  df-cn 23096  df-cnp 23097  df-tx 23431  df-tmd 23941

This theorem is referenced by:  tsmsxp  24024