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Theorem tmdcn2 22113
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 22105 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43ad2antrr 705 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5 eqid 2771 . . . . . 6 (+𝑓𝐺) = (+𝑓𝐺)
61, 5tmdcn 22107 . . . . 5 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
76ad2antrr 705 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8 simpr1 1233 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋𝐵)
9 simpr2 1235 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌𝐵)
10 opelxpi 5288 . . . . . 6 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
118, 9, 10syl2anc 573 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
12 txtopon 21615 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
134, 4, 12syl2anc 573 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
14 toponuni 20939 . . . . . 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 2852 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽))
17 eqid 2771 . . . . 5 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
1817cncnpi 21303 . . . 4 (((+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
197, 16, 18syl2anc 573 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘⟨𝑋, 𝑌⟩))
20 simplr 752 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑈𝐽)
21 tmdcn2.3 . . . . . 6 + = (+g𝐺)
222, 21, 5plusfval 17456 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
238, 9, 22syl2anc 573 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) = (𝑋 + 𝑌))
24 simpr3 1237 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
2523, 24eqeltrd 2850 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) ∈ 𝑈)
264, 4, 19, 20, 8, 9, 25txcnpi 21632 . 2 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)))
27 dfss3 3741 . . . . . . 7 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈))
28 eleq1 2838 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈)))
292, 5plusffn 17458 . . . . . . . . . 10 (+𝑓𝐺) Fn (𝐵 × 𝐵)
30 elpreima 6480 . . . . . . . . . 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 276 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
3332ralxp 5402 . . . . . . 7 (∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
3427, 33bitri 264 . . . . . 6 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
35 opelxp 5286 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵))
36 df-ov 6796 . . . . . . . . . . 11 (𝑥(+𝑓𝐺)𝑦) = ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩)
372, 21, 5plusfval 17456 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝑥(+𝑓𝐺)𝑦) = (𝑥 + 𝑦))
3836, 37syl5eqr 2819 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3935, 38sylbi 207 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
4039eleq1d 2835 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
4140biimpa 462 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
42412ralimi 3102 . . . . . 6 (∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
4334, 42sylbi 207 . . . . 5 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
44433anim3i 1157 . . . 4 ((𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4544reximi 3159 . . 3 (∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4645reximi 3159 . 2 (∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4726, 46syl 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723  cop 4322   cuni 4574   × cxp 5247  ccnv 5248  cima 5252   Fn wfn 6026  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  TopOpenctopn 16290  +𝑓cplusf 17447  TopOnctopon 20935   Cn ccn 21249   CnP ccnp 21250   ×t ctx 21584  TopMndctmd 22094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-topgen 16312  df-plusf 17449  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cn 21252  df-cnp 21253  df-tx 21586  df-tmd 22096
This theorem is referenced by:  tsmsxp  22178
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