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Theorem cnmpt1plusg 24007
Description: Continuity of the group sum; analogue of cnmpt12f 23586 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
tgpcn.j 𝐽 = (TopOpenβ€˜πΊ)
cnmpt1plusg.p + = (+gβ€˜πΊ)
cnmpt1plusg.g (πœ‘ β†’ 𝐺 ∈ TopMnd)
cnmpt1plusg.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))
cnmpt1plusg.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
cnmpt1plusg.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐾 Cn 𝐽))
Assertion
Ref Expression
cnmpt1plusg (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 + 𝐡)) ∈ (𝐾 Cn 𝐽))
Distinct variable groups:   π‘₯,𝐺   π‘₯,𝐽   π‘₯,𝐾   πœ‘,π‘₯   π‘₯,𝑋
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘₯)   + (π‘₯)

Proof of Theorem cnmpt1plusg
StepHypRef Expression
1 cnmpt1plusg.k . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))
2 cnmpt1plusg.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TopMnd)
3 tgpcn.j . . . . . . . 8 𝐽 = (TopOpenβ€˜πΊ)
4 eqid 2725 . . . . . . . 8 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
53, 4tmdtopon 24001 . . . . . . 7 (𝐺 ∈ TopMnd β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
62, 5syl 17 . . . . . 6 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
7 cnmpt1plusg.a . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
8 cnf2 23169 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆ(Baseβ€˜πΊ))
91, 6, 7, 8syl3anc 1368 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆ(Baseβ€˜πΊ))
109fvmptelcdm 7117 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ (Baseβ€˜πΊ))
11 cnmpt1plusg.b . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐾 Cn 𝐽))
12 cnf2 23169 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐾 Cn 𝐽)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆ(Baseβ€˜πΊ))
131, 6, 11, 12syl3anc 1368 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆ(Baseβ€˜πΊ))
1413fvmptelcdm 7117 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ (Baseβ€˜πΊ))
15 cnmpt1plusg.p . . . . 5 + = (+gβ€˜πΊ)
16 eqid 2725 . . . . 5 (+π‘“β€˜πΊ) = (+π‘“β€˜πΊ)
174, 15, 16plusfval 18604 . . . 4 ((𝐴 ∈ (Baseβ€˜πΊ) ∧ 𝐡 ∈ (Baseβ€˜πΊ)) β†’ (𝐴(+π‘“β€˜πΊ)𝐡) = (𝐴 + 𝐡))
1810, 14, 17syl2anc 582 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝐴(+π‘“β€˜πΊ)𝐡) = (𝐴 + 𝐡))
1918mpteq2dva 5243 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴(+π‘“β€˜πΊ)𝐡)) = (π‘₯ ∈ 𝑋 ↦ (𝐴 + 𝐡)))
203, 16tmdcn 24003 . . . 4 (𝐺 ∈ TopMnd β†’ (+π‘“β€˜πΊ) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
212, 20syl 17 . . 3 (πœ‘ β†’ (+π‘“β€˜πΊ) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
221, 7, 11, 21cnmpt12f 23586 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴(+π‘“β€˜πΊ)𝐡)) ∈ (𝐾 Cn 𝐽))
2319, 22eqeltrrd 2826 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 + 𝐡)) ∈ (𝐾 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5226  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415  Basecbs 17177  +gcplusg 17230  TopOpenctopn 17400  +𝑓cplusf 18594  TopOnctopon 22828   Cn ccn 23144   Γ—t ctx 23480  TopMndctmd 23990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-map 8843  df-topgen 17422  df-plusf 18596  df-top 22812  df-topon 22829  df-topsp 22851  df-bases 22865  df-cn 23147  df-tx 23482  df-tmd 23992
This theorem is referenced by:  tmdmulg  24012  tmdgsum  24015  tmdlactcn  24022  clsnsg  24030  tgpt0  24039  cnmpt1mulr  24102
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