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Mirrors > Home > MPE Home > Th. List > gsumcllem | Structured version Visualization version GIF version |
Description: Lemma for gsumcl 19700 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 6914 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
3 | 2 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
4 | difeq2 4080 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅)) | |
5 | dif0 4336 | . . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
6 | 4, 5 | eqtrdi 2789 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴) |
7 | 6 | eleq2d 2820 | . . . . . 6 ⊢ (𝑊 = ∅ → (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑘 ∈ 𝐴)) |
8 | 7 | biimpar 479 | . . . . 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 8131 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
13 | 8, 12 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑘) = 𝑍) |
14 | 13 | anassrs 469 | . . 3 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = 𝑍) |
15 | 14 | mpteq2dva 5209 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
16 | 3, 15 | eqtrd 2773 | 1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3911 ⊆ wss 3914 ∅c0 4286 ↦ cmpt 5192 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 supp csupp 8096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-supp 8097 |
This theorem is referenced by: gsumzres 19694 gsumzcl2 19695 gsumzf1o 19697 gsumzaddlem 19706 gsumzmhm 19722 gsumzoppg 19729 |
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