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Theorem tmdcn2 21703
 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 21695 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43ad2antrr 758 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5 eqid 2610 . . . . . 6 (+𝑓𝐺) = (+𝑓𝐺)
61, 5tmdcn 21697 . . . . 5 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
76ad2antrr 758 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8 simpr1 1060 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋𝐵)
9 simpr2 1061 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌𝐵)
10 opelxpi 5072 . . . . . 6 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
118, 9, 10syl2anc 691 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
12 txtopon 21204 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
134, 4, 12syl2anc 691 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
14 toponuni 20542 . . . . . 6 ((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) → (𝐵 × 𝐵) = (𝐽 ×t 𝐽))
1513, 14syl 17 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐵 × 𝐵) = (𝐽 ×t 𝐽))
1611, 15eleqtrd 2690 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽))
17 eqid 2610 . . . . 5 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
1817cncnpi 20892 . . . 4 (((+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
197, 16, 18syl2anc 691 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
20 simplr 788 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑈𝐽)
21 tmdcn2.3 . . . . . 6 + = (+g𝐺)
222, 21, 5plusfval 17071 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
238, 9, 22syl2anc 691 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
24 simpr3 1062 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
2523, 24eqeltrd 2688 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) ∈ 𝑈)
264, 4, 19, 20, 8, 9, 25txcnpi 21221 . 2 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)))
27 dfss3 3558 . . . . . . 7 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈))
28 eleq1 2676 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈)))
292, 5plusffn 17073 . . . . . . . . . 10 (+𝑓𝐺) Fn (𝐵 × 𝐵)
30 elpreima 6245 . . . . . . . . . 10 ((+𝑓𝐺) Fn (𝐵 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
3129, 30ax-mp 5 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
3228, 31syl6bb 275 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
3332ralxp 5185 . . . . . . 7 (∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
3427, 33bitri 263 . . . . . 6 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
35 opelxp 5070 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵))
36 df-ov 6552 . . . . . . . . . . . 12 (𝑥(+𝑓𝐺)𝑦) = ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩)
372, 21, 5plusfval 17071 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐵) → (𝑥(+𝑓𝐺)𝑦) = (𝑥 + 𝑦))
3836, 37syl5eqr 2658 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3935, 38sylbi 206 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
4039eleq1d 2672 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
4140biimpa 500 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
4241ralimi 2936 . . . . . . 7 (∀𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → ∀𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
4342ralimi 2936 . . . . . 6 (∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
4434, 43sylbi 206 . . . . 5 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
45443anim3i 1243 . . . 4 ((𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4645reximi 2994 . . 3 (∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4746reximi 2994 . 2 (∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4826, 47syl 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372   × cxp 5036  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  +𝑓cplusf 17062  TopOnctopon 20518   Cn ccn 20838   CnP ccnp 20839   ×t ctx 21173  TopMndctmd 21684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-plusf 17064  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-tx 21175  df-tmd 21686 This theorem is referenced by:  tsmsxp  21768
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