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Theorem tmdcn2 24113
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 + = (+g𝐺)
Assertion
Ref Expression
tmdcn2 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (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.
StepHypRef Expression
1 tmdcn2.2 . . . . 5 𝐽 = (TopOpen‘𝐺)
2 tmdcn2.1 . . . . 5 𝐵 = (Base‘𝐺)
31, 2tmdtopon 24105 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43ad2antrr 726 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5 eqid 2735 . . . . . 6 (+𝑓𝐺) = (+𝑓𝐺)
61, 5tmdcn 24107 . . . . 5 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
76ad2antrr 726 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8 simpr1 1193 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋𝐵)
9 simpr2 1194 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌𝐵)
108, 9opelxpd 5728 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 txtopon 23615 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
124, 4, 11syl2anc 584 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
13 toponuni 22936 . . . . . 6 ((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) → (𝐵 × 𝐵) = (𝐽 ×t 𝐽))
1412, 13syl 17 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐵 × 𝐵) = (𝐽 ×t 𝐽))
1510, 14eleqtrd 2841 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽))
16 eqid 2735 . . . . 5 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
1716cncnpi 23302 . . . 4 (((+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
187, 15, 17syl2anc 584 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
19 simplr 769 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑈𝐽)
20 tmdcn2.3 . . . . . 6 + = (+g𝐺)
212, 20, 5plusfval 18673 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
228, 9, 21syl2anc 584 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
23 simpr3 1195 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
2422, 23eqeltrd 2839 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) ∈ 𝑈)
254, 4, 18, 19, 8, 9, 24txcnpi 23632 . 2 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)))
26 dfss3 3984 . . . . . . 7 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈))
27 eleq1 2827 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈)))
282, 5plusffn 18675 . . . . . . . . . 10 (+𝑓𝐺) Fn (𝐵 × 𝐵)
29 elpreima 7078 . . . . . . . . . 10 ((+𝑓𝐺) Fn (𝐵 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
3028, 29ax-mp 5 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
3127, 30bitrdi 287 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
3231ralxp 5855 . . . . . . 7 (∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
3326, 32bitri 275 . . . . . 6 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
34 opelxp 5725 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵))
35 df-ov 7434 . . . . . . . . . . 11 (𝑥(+𝑓𝐺)𝑦) = ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩)
362, 20, 5plusfval 18673 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝑥(+𝑓𝐺)𝑦) = (𝑥 + 𝑦))
3735, 36eqtr3id 2789 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3834, 37sylbi 217 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3938eleq1d 2824 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
4039biimpa 476 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
41402ralimi 3121 . . . . . 6 (∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
4233, 41sylbi 217 . . . . 5 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
43423anim3i 1153 . . . 4 ((𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4443reximi 3082 . . 3 (∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4544reximi 3082 . 2 (∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4625, 45syl 17 1 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  cop 4637   cuni 4912   × cxp 5687  ccnv 5688  cima 5692   Fn wfn 6558  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  +𝑓cplusf 18663  TopOnctopon 22932   Cn ccn 23248   CnP ccnp 23249   ×t ctx 23584  TopMndctmd 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-plusf 18665  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-tx 23586  df-tmd 24096
This theorem is referenced by:  tsmsxp  24179
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