Theorem tmdcn2 24316

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 24308	. . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4	3	ad2antrr 739	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5		eqid 2760	. . . . . 6 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
6	1, 5	tmdcn 24310	. . . . 5 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
7	6	ad2antrr 739	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8		simpr1 1213	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋 ∈ 𝐵)
9		simpr2 1214	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌 ∈ 𝐵)
10	8, 9	opelxpd 5694	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		txtopon 23818	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
12	4, 4, 11	syl2anc 596	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
13		toponuni 23140	. . . . . 6 ⊢ ((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) → (𝐵 × 𝐵) = ∪ (𝐽 ×t 𝐽))
14	12, 13	syl 18	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐵 × 𝐵) = ∪ (𝐽 ×t 𝐽))
15	10, 14	eleqtrd 2862	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ ∪ (𝐽 ×t 𝐽))
16		eqid 2760	. . . . 5 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
17	16	cncnpi 23504	. . . 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 596	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓‘𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
19		simplr 781	. . 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 18738	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+𝑓‘𝐺)𝑌) = (𝑋 + 𝑌))
22	8, 9, 21	syl2anc 596	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓‘𝐺)𝑌) = (𝑋 + 𝑌))
23		simpr3 1215	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
24	22, 23	eqeltrd 2860	. . 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 23835	. 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 2848	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡(+𝑓‘𝐺) “ 𝑈)))
28	2, 5	plusffn 18740	. . . . . . . . . 10 ⊢ (+𝑓‘𝐺) Fn (𝐵 × 𝐵)
29		elpreima 7051	. . . . . . . . . 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 290	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
32	31	ralxp 5821	. . . . . . 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 278	. . . . . 6 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
34		opelxp 5691	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
35		df-ov 7417	. . . . . . . . . . 11 ⊢ (𝑥(+𝑓‘𝐺)𝑦) = ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩)
36	2, 20, 5	plusfval 18738	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+𝑓‘𝐺)𝑦) = (𝑥 + 𝑦))
37	35, 36	eqtr3id 2809	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
38	34, 37	sylbi 220	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
39	38	eleq1d 2845	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
40	39	biimpa 482	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
41	40	2ralimi 3132	. . . . . 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 220	. . . . 5 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) → ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈)
43	42	3anim3i 1172	. . . 4 ⊢ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈)) → (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))
44	43	reximi 3100	. . 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 3100	. 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 18	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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ⟨cop 4590  ∪ cuni 4867   × cxp 5653  ^◡ccnv 5654   “ cima 5658   Fn wfn 6528  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  +_gcplusg 17343  TopOpenctopn 17507  +_𝑓cplusf 18728  TopOnctopon 23136   Cn ccn 23450   CnP ccnp 23451   ×_t ctx 23787  TopMndctmd 24297

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8829  df-topgen 17529  df-plusf 18730  df-top 23120  df-topon 23137  df-topsp 23159  df-bases 23172  df-cn 23453  df-cnp 23454  df-tx 23789  df-tmd 24299

This theorem is used by:  tsmsxp  24382