Theorem tmdcn2 23983

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 23975	. . . 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 2730	. . . . . 6 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
6	1, 5	tmdcn 23977	. . . . 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 5680	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		txtopon 23485	. . . . . . 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 22808	. . . . . 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 2831	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ ∪ (𝐽 ×t 𝐽))
16		eqid 2730	. . . . 5 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
17	16	cncnpi 23172	. . . 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 18581	. . . . 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 2829	. . 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 23502	. 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 3938	. . . . . . 7 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈))
27		eleq1 2817	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡(+𝑓‘𝐺) “ 𝑈)))
28	2, 5	plusffn 18583	. . . . . . . . . 10 ⊢ (+𝑓‘𝐺) Fn (𝐵 × 𝐵)
29		elpreima 7033	. . . . . . . . . 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 5808	. . . . . . 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 5677	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
35		df-ov 7393	. . . . . . . . . . 11 ⊢ (𝑥(+𝑓‘𝐺)𝑦) = ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩)
36	2, 20, 5	plusfval 18581	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+𝑓‘𝐺)𝑦) = (𝑥 + 𝑦))
37	35, 36	eqtr3id 2779	. . . . . . . . . 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 2814	. . . . . . . 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 3104	. . . . . 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 3068	. . 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 3068	. 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 3045  ∃wrex 3054   ⊆ wss 3917  ⟨cop 4598  ∪ cuni 4874   × cxp 5639  ^◡ccnv 5640   “ cima 5644   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  TopOpenctopn 17391  +_𝑓cplusf 18571  TopOnctopon 22804   Cn ccn 23118   CnP ccnp 23119   ×_t ctx 23454  TopMndctmd 23964

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-topgen 17413  df-plusf 18573  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cn 23121  df-cnp 23122  df-tx 23456  df-tmd 23966

This theorem is referenced by:  tsmsxp  24049