Theorem gsumcllem 19806

Description: Lemma for gsumcl 19813 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 6895	. . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
4		difeq2 4073	. . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅))
5		dif0 4331	. . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴
6	4, 5	eqtrdi 2780	. . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴)
7	6	eleq2d 2814	. . . . . 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 8135	. . . . 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 5188	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍))
16	3, 15	eqtrd 2764	1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286   ↦ cmpt 5176  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353   supp csupp 8100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-supp 8101

This theorem is referenced by:  gsumzres  19807  gsumzcl2  19808  gsumzf1o  19810  gsumzaddlem  19819  gsumzmhm  19835  gsumzoppg  19842