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Mirrors > Home > MPE Home > Th. List > gsumcllem | Structured version Visualization version GIF version |
Description: Lemma for gsumcl 19869 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 6967 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
4 | difeq2 4114 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅)) | |
5 | dif0 4373 | . . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
6 | 4, 5 | eqtrdi 2784 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴) |
7 | 6 | eleq2d 2815 | . . . . . 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 8200 | . . . . 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 5248 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
16 | 3, 15 | eqtrd 2768 | 1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4323 ↦ cmpt 5231 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 supp csupp 8165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-supp 8166 |
This theorem is referenced by: gsumzres 19863 gsumzcl2 19864 gsumzf1o 19866 gsumzaddlem 19875 gsumzmhm 19891 gsumzoppg 19898 |
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