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| Mirrors > Home > MPE Home > Th. List > gsumcllem | Structured version Visualization version GIF version | ||
| Description: Lemma for gsumcl 19933 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.) |
| 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) |
| Ref | Expression |
|---|---|
| gsumcllem | ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
| 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 4120 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅)) | |
| 5 | dif0 4378 | . . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
| 6 | 4, 5 | eqtrdi 2793 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴) |
| 7 | 6 | eleq2d 2827 | . . . . . 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 8220 | . . . . 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 5242 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
| 16 | 3, 15 | eqtrd 2777 | 1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8186 |
| This theorem is referenced by: gsumzres 19927 gsumzcl2 19928 gsumzf1o 19930 gsumzaddlem 19939 gsumzmhm 19955 gsumzoppg 19962 |
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