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Theorem gsumcllem 19824
Description: Lemma for gsumcl 19831 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumcllem.f (𝜑𝐹:𝐴𝐵)
gsumcllem.a (𝜑𝐴𝑉)
gsumcllem.z (𝜑𝑍𝑈)
gsumcllem.s (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
Assertion
Ref Expression
gsumcllem ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴𝑍))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝜑,𝑘   𝑘,𝑊
Allowed substitution hints:   𝐵(𝑘)   𝑈(𝑘)   𝑉(𝑘)   𝑍(𝑘)

Proof of Theorem gsumcllem
StepHypRef Expression
1 gsumcllem.f . . . 4 (𝜑𝐹:𝐴𝐵)
21feqmptd 6951 . . 3 (𝜑𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
32adantr 480 . 2 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
4 difeq2 4109 . . . . . . . 8 (𝑊 = ∅ → (𝐴𝑊) = (𝐴 ∖ ∅))
5 dif0 4365 . . . . . . . 8 (𝐴 ∖ ∅) = 𝐴
64, 5eqtrdi 2780 . . . . . . 7 (𝑊 = ∅ → (𝐴𝑊) = 𝐴)
76eleq2d 2811 . . . . . 6 (𝑊 = ∅ → (𝑘 ∈ (𝐴𝑊) ↔ 𝑘𝐴))
87biimpar 477 . . . . 5 ((𝑊 = ∅ ∧ 𝑘𝐴) → 𝑘 ∈ (𝐴𝑊))
9 gsumcllem.s . . . . . 6 (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
10 gsumcllem.a . . . . . 6 (𝜑𝐴𝑉)
11 gsumcllem.z . . . . . 6 (𝜑𝑍𝑈)
121, 9, 10, 11suppssr 8176 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → (𝐹𝑘) = 𝑍)
138, 12sylan2 592 . . . 4 ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘𝐴)) → (𝐹𝑘) = 𝑍)
1413anassrs 467 . . 3 (((𝜑𝑊 = ∅) ∧ 𝑘𝐴) → (𝐹𝑘) = 𝑍)
1514mpteq2dva 5239 . 2 ((𝜑𝑊 = ∅) → (𝑘𝐴 ↦ (𝐹𝑘)) = (𝑘𝐴𝑍))
163, 15eqtrd 2764 1 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cdif 3938  wss 3941  c0 4315  cmpt 5222  wf 6530  cfv 6534  (class class class)co 7402   supp csupp 8141
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-supp 8142
This theorem is referenced by:  gsumzres  19825  gsumzcl2  19826  gsumzf1o  19828  gsumzaddlem  19837  gsumzmhm  19853  gsumzoppg  19860
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