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Theorem gsumcllem 19693
Description: Lemma for gsumcl 19700 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 6914 . . 3 (𝜑𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
32adantr 482 . 2 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
4 difeq2 4080 . . . . . . . 8 (𝑊 = ∅ → (𝐴𝑊) = (𝐴 ∖ ∅))
5 dif0 4336 . . . . . . . 8 (𝐴 ∖ ∅) = 𝐴
64, 5eqtrdi 2789 . . . . . . 7 (𝑊 = ∅ → (𝐴𝑊) = 𝐴)
76eleq2d 2820 . . . . . 6 (𝑊 = ∅ → (𝑘 ∈ (𝐴𝑊) ↔ 𝑘𝐴))
87biimpar 479 . . . . 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 8131 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → (𝐹𝑘) = 𝑍)
138, 12sylan2 594 . . . 4 ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘𝐴)) → (𝐹𝑘) = 𝑍)
1413anassrs 469 . . 3 (((𝜑𝑊 = ∅) ∧ 𝑘𝐴) → (𝐹𝑘) = 𝑍)
1514mpteq2dva 5209 . 2 ((𝜑𝑊 = ∅) → (𝑘𝐴 ↦ (𝐹𝑘)) = (𝑘𝐴𝑍))
163, 15eqtrd 2773 1 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cdif 3911  wss 3914  c0 4286  cmpt 5192  wf 6496  cfv 6500  (class class class)co 7361   supp csupp 8096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-supp 8097
This theorem is referenced by:  gsumzres  19694  gsumzcl2  19695  gsumzf1o  19697  gsumzaddlem  19706  gsumzmhm  19722  gsumzoppg  19729
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