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Theorem tmdcn2 22697
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 22689 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43ad2antrr 725 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5 eqid 2824 . . . . . 6 (+𝑓𝐺) = (+𝑓𝐺)
61, 5tmdcn 22691 . . . . 5 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
76ad2antrr 725 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8 simpr1 1191 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋𝐵)
9 simpr2 1192 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌𝐵)
108, 9opelxpd 5580 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 txtopon 22199 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
124, 4, 11syl2anc 587 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
13 toponuni 21522 . . . . . 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 2918 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽))
16 eqid 2824 . . . . 5 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
1716cncnpi 21886 . . . 4 (((+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
187, 15, 17syl2anc 587 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
19 simplr 768 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑈𝐽)
20 tmdcn2.3 . . . . . 6 + = (+g𝐺)
212, 20, 5plusfval 17859 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
228, 9, 21syl2anc 587 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
23 simpr3 1193 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
2422, 23eqeltrd 2916 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) ∈ 𝑈)
254, 4, 18, 19, 8, 9, 24txcnpi 22216 . 2 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)))
26 dfss3 3941 . . . . . . 7 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈))
27 eleq1 2903 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈)))
282, 5plusffn 17861 . . . . . . . . . 10 (+𝑓𝐺) Fn (𝐵 × 𝐵)
29 elpreima 6819 . . . . . . . . . 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, 30syl6bb 290 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
3231ralxp 5699 . . . . . . 7 (∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
3326, 32bitri 278 . . . . . 6 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
34 opelxp 5578 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵))
35 df-ov 7152 . . . . . . . . . . 11 (𝑥(+𝑓𝐺)𝑦) = ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩)
362, 20, 5plusfval 17859 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝑥(+𝑓𝐺)𝑦) = (𝑥 + 𝑦))
3735, 36syl5eqr 2873 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3834, 37sylbi 220 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3938eleq1d 2900 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
4039biimpa 480 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
41402ralimi 3156 . . . . . 6 (∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
4233, 41sylbi 220 . . . . 5 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
43423anim3i 1151 . . . 4 ((𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4443reximi 3237 . . 3 (∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4544reximi 3237 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  wss 3919  cop 4556   cuni 4824   × cxp 5540  ccnv 5541  cima 5545   Fn wfn 6338  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  TopOpenctopn 16695  +𝑓cplusf 17849  TopOnctopon 21518   Cn ccn 21832   CnP ccnp 21833   ×t ctx 22168  TopMndctmd 22678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-topgen 16717  df-plusf 17851  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cn 21835  df-cnp 21836  df-tx 22170  df-tmd 22680
This theorem is referenced by:  tsmsxp  22763
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