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Mirrors > Home > MPE Home > Th. List > gsumcllem | Structured version Visualization version GIF version |
Description: Lemma for gsumcl 19028 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 6708 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
3 | 2 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
4 | difeq2 4044 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅)) | |
5 | dif0 4286 | . . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
6 | 4, 5 | eqtrdi 2849 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴) |
7 | 6 | eleq2d 2875 | . . . . . 6 ⊢ (𝑊 = ∅ → (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑘 ∈ 𝐴)) |
8 | 7 | biimpar 481 | . . . . 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 7844 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
13 | 8, 12 | sylan2 595 | . . . 4 ⊢ ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑘) = 𝑍) |
14 | 13 | anassrs 471 | . . 3 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = 𝑍) |
15 | 14 | mpteq2dva 5125 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
16 | 3, 15 | eqtrd 2833 | 1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supp csupp 7813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-supp 7814 |
This theorem is referenced by: gsumzres 19022 gsumzcl2 19023 gsumzf1o 19025 gsumzaddlem 19034 gsumzmhm 19050 gsumzoppg 19057 |
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