Theorem gsumcllem 19950

Description: Lemma for gsumcl 19957 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 6990	. . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
4		difeq2 4143	. . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅))
5		dif0 4400	. . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴
6	4, 5	eqtrdi 2796	. . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴)
7	6	eleq2d 2830	. . . . . 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 8236	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)
13	8, 12	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑘) = 𝑍)
14	13	anassrs 467	. . 3 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = 𝑍)
15	14	mpteq2dva 5266	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍))
16	3, 15	eqtrd 2780	1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   supp csupp 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-supp 8202

This theorem is referenced by:  gsumzres  19951  gsumzcl2  19952  gsumzf1o  19954  gsumzaddlem  19963  gsumzmhm  19979  gsumzoppg  19986