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Theorem gsumcllem 19020
Description: Lemma for gsumcl 19027 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 6726 . . 3 (𝜑𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
32adantr 483 . 2 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
4 difeq2 4091 . . . . . . . 8 (𝑊 = ∅ → (𝐴𝑊) = (𝐴 ∖ ∅))
5 dif0 4330 . . . . . . . 8 (𝐴 ∖ ∅) = 𝐴
64, 5syl6eq 2870 . . . . . . 7 (𝑊 = ∅ → (𝐴𝑊) = 𝐴)
76eleq2d 2896 . . . . . 6 (𝑊 = ∅ → (𝑘 ∈ (𝐴𝑊) ↔ 𝑘𝐴))
87biimpar 480 . . . . 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 7853 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → (𝐹𝑘) = 𝑍)
138, 12sylan2 594 . . . 4 ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘𝐴)) → (𝐹𝑘) = 𝑍)
1413anassrs 470 . . 3 (((𝜑𝑊 = ∅) ∧ 𝑘𝐴) → (𝐹𝑘) = 𝑍)
1514mpteq2dva 5152 . 2 ((𝜑𝑊 = ∅) → (𝑘𝐴 ↦ (𝐹𝑘)) = (𝑘𝐴𝑍))
163, 15eqtrd 2854 1 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  cdif 3931  wss 3934  c0 4289  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7148   supp csupp 7822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-supp 7823
This theorem is referenced by:  gsumzres  19021  gsumzcl2  19022  gsumzf1o  19024  gsumzaddlem  19033  gsumzmhm  19049  gsumzoppg  19056
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