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Theorem gsumcllem 18233
Description: Lemma for gsumcl 18240 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 6208 . . 3 (𝜑𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
32adantr 481 . 2 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
4 difeq2 3702 . . . . . . . 8 (𝑊 = ∅ → (𝐴𝑊) = (𝐴 ∖ ∅))
5 dif0 3926 . . . . . . . 8 (𝐴 ∖ ∅) = 𝐴
64, 5syl6eq 2671 . . . . . . 7 (𝑊 = ∅ → (𝐴𝑊) = 𝐴)
76eleq2d 2684 . . . . . 6 (𝑊 = ∅ → (𝑘 ∈ (𝐴𝑊) ↔ 𝑘𝐴))
87biimpar 502 . . . . 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 7274 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → (𝐹𝑘) = 𝑍)
138, 12sylan2 491 . . . 4 ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘𝐴)) → (𝐹𝑘) = 𝑍)
1413anassrs 679 . . 3 (((𝜑𝑊 = ∅) ∧ 𝑘𝐴) → (𝐹𝑘) = 𝑍)
1514mpteq2dva 4706 . 2 ((𝜑𝑊 = ∅) → (𝑘𝐴 ↦ (𝐹𝑘)) = (𝑘𝐴𝑍))
163, 15eqtrd 2655 1 ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cdif 3553  wss 3556  c0 3893  cmpt 4675  wf 5845  cfv 5849  (class class class)co 6607   supp csupp 7243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-supp 7244
This theorem is referenced by:  gsumzres  18234  gsumzcl2  18235  gsumzf1o  18237  gsumzaddlem  18245  gsumzmhm  18261  gsumzoppg  18268
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