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Theorem tmdcn2 23440
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 23432 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43ad2antrr 724 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝐽 ∈ (TopOn‘𝐵))
5 eqid 2736 . . . . . 6 (+𝑓𝐺) = (+𝑓𝐺)
61, 5tmdcn 23434 . . . . 5 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
76ad2antrr 724 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
8 simpr1 1194 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑋𝐵)
9 simpr2 1195 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑌𝐵)
108, 9opelxpd 5671 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 txtopon 22942 . . . . . . 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 22263 . . . . . 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 2840 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ⟨𝑋, 𝑌⟩ ∈ (𝐽 ×t 𝐽))
16 eqid 2736 . . . . 5 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
1716cncnpi 22629 . . . 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 767 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → 𝑈𝐽)
20 tmdcn2.3 . . . . . 6 + = (+g𝐺)
212, 20, 5plusfval 18504 . . . . 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 1196 . . . 4 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)
2422, 23eqeltrd 2838 . . 3 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → (𝑋(+𝑓𝐺)𝑌) ∈ 𝑈)
254, 4, 18, 19, 8, 9, 24txcnpi 22959 . 2 (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)))
26 dfss3 3932 . . . . . . 7 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈))
27 eleq1 2825 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ⟨𝑥, 𝑦⟩ ∈ ((+𝑓𝐺) “ 𝑈)))
282, 5plusffn 18506 . . . . . . . . . 10 (+𝑓𝐺) Fn (𝐵 × 𝐵)
29 elpreima 7008 . . . . . . . . . 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 286 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈)))
3231ralxp 5797 . . . . . . 7 (∀𝑧 ∈ (𝑢 × 𝑣)𝑧 ∈ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
3326, 32bitri 274 . . . . . 6 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) ↔ ∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈))
34 opelxp 5669 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵))
35 df-ov 7360 . . . . . . . . . . 11 (𝑥(+𝑓𝐺)𝑦) = ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩)
362, 20, 5plusfval 18504 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝑥(+𝑓𝐺)𝑦) = (𝑥 + 𝑦))
3735, 36eqtr3id 2790 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3834, 37sylbi 216 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) = (𝑥 + 𝑦))
3938eleq1d 2822 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → (((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈 ↔ (𝑥 + 𝑦) ∈ 𝑈))
4039biimpa 477 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → (𝑥 + 𝑦) ∈ 𝑈)
41402ralimi 3126 . . . . . 6 (∀𝑥𝑢𝑦𝑣 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ∧ ((+𝑓𝐺)‘⟨𝑥, 𝑦⟩) ∈ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
4233, 41sylbi 216 . . . . 5 ((𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈) → ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈)
43423anim3i 1154 . . . 4 ((𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4443reximi 3087 . . 3 (∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ (𝑢 × 𝑣) ⊆ ((+𝑓𝐺) “ 𝑈)) → ∃𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))
4544reximi 3087 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  wss 3910  cop 4592   cuni 4865   × cxp 5631  ccnv 5632  cima 5636   Fn wfn 6491  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  TopOpenctopn 17303  +𝑓cplusf 18494  TopOnctopon 22259   Cn ccn 22575   CnP ccnp 22576   ×t ctx 22911  TopMndctmd 23421
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-topgen 17325  df-plusf 18496  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cn 22578  df-cnp 22579  df-tx 22913  df-tmd 23423
This theorem is referenced by:  tsmsxp  23506
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