Theorem tmdcn2 24075

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 24067	. . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4	3	ad2antrr 733	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5		eqid 2741	. . . . . 6 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
6	1, 5	tmdcn 24069	. . . . 5 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
7	6	ad2antrr 733	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8		simpr1 1202	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋 ∈ 𝐵)
9		simpr2 1203	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌 ∈ 𝐵)
10	8, 9	opelxpd 5659	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		txtopon 23577	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
12	4, 4, 11	syl2anc 591	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
13		toponuni 22900	. . . . . 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 2843	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ ∪ (𝐽 ×t 𝐽))
16		eqid 2741	. . . . 5 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
17	16	cncnpi 23264	. . . 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 591	. . 3 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓‘𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
19		simplr 775	. . 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 18610	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+𝑓‘𝐺)𝑌) = (𝑋 + 𝑌))
22	8, 9, 21	syl2anc 591	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓‘𝐺)𝑌) = (𝑋 + 𝑌))
23		simpr3 1204	. . . 4 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
24	22, 23	eqeltrd 2841	. . 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 23594	. 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 3905	. . . . . . 7 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈))
27		eleq1 2829	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡(+𝑓‘𝐺) “ 𝑈)))
28	2, 5	plusffn 18612	. . . . . . . . . 10 ⊢ (+𝑓‘𝐺) Fn (𝐵 × 𝐵)
29		elpreima 7002	. . . . . . . . . 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 289	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (◡(+𝑓‘𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
32	31	ralxp 5785	. . . . . . 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 277	. . . . . 6 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) ↔ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
34		opelxp 5656	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
35		df-ov 7362	. . . . . . . . . . 11 ⊢ (𝑥(+𝑓‘𝐺)𝑦) = ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩)
36	2, 20, 5	plusfval 18610	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+𝑓‘𝐺)𝑦) = (𝑥 + 𝑦))
37	35, 36	eqtr3id 2790	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
38	34, 37	sylbi 219	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
39	38	eleq1d 2826	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
40	39	biimpa 478	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
41	40	2ralimi 3111	. . . . . 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 219	. . . . 5 ⊢ ((𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈) → ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈)
43	42	3anim3i 1161	. . . 4 ⊢ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡(+𝑓‘𝐺) “ 𝑈)) → (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))
44	43	reximi 3079	. . 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 3079	. 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 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ⊆ wss 3884  ⟨cop 4563  ∪ cuni 4840   × cxp 5618  ^◡ccnv 5619   “ cima 5623   Fn wfn 6483  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  +_gcplusg 17215  TopOpenctopn 17379  +_𝑓cplusf 18600  TopOnctopon 22896   Cn ccn 23210   CnP ccnp 23211   ×_t ctx 23546  TopMndctmd 24056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-map 8769  df-topgen 17401  df-plusf 18602  df-top 22880  df-topon 22897  df-topsp 22919  df-bases 22932  df-cn 23213  df-cnp 23214  df-tx 23548  df-tmd 24058

This theorem is referenced by:  tsmsxp  24141