Theorem gsumcllem 19941

Description: Lemma for gsumcl 19948 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	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumcllem.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumcllem.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
gsumcllem.s	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)

Assertion

Ref	Expression
gsumcllem	⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍))

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

Step	Hyp	Ref	Expression
1		gsumcllem.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	feqmptd 6977	. . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
4		difeq2 4130	. . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅))
5		dif0 4384	. . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴
6	4, 5	eqtrdi 2791	. . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴)
7	6	eleq2d 2825	. . . . . 6 ⊢ (𝑊 = ∅ → (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑘 ∈ 𝐴))
8	7	biimpar 477	. . . . 5 ⊢ ((𝑊 = ∅ ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝐴 ∖ 𝑊))
9		gsumcllem.s	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
10		gsumcllem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
11		gsumcllem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
12	1, 9, 10, 11	suppssr 8219	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)
13	8, 12	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑘) = 𝑍)
14	13	anassrs 467	. . 3 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = 𝑍)
15	14	mpteq2dva 5248	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍))
16	3, 15	eqtrd 2775	1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339   ↦ cmpt 5231  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   supp csupp 8184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185

This theorem is referenced by:  gsumzres  19942  gsumzcl2  19943  gsumzf1o  19945  gsumzaddlem  19954  gsumzmhm  19970  gsumzoppg  19977