Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > gsumcllem | Structured version Visualization version GIF version |
Description: Lemma for gsumcl 18316 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 6249 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
3 | 2 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
4 | difeq2 3722 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = (𝐴 ∖ ∅)) | |
5 | dif0 3950 | . . . . . . . 8 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
6 | 4, 5 | syl6eq 2672 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝐴 ∖ 𝑊) = 𝐴) |
7 | 6 | eleq2d 2687 | . . . . . 6 ⊢ (𝑊 = ∅ → (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑘 ∈ 𝐴)) |
8 | 7 | biimpar 502 | . . . . 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 7326 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
13 | 8, 12 | sylan2 491 | . . . 4 ⊢ ((𝜑 ∧ (𝑊 = ∅ ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑘) = 𝑍) |
14 | 13 | anassrs 680 | . . 3 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = 𝑍) |
15 | 14 | mpteq2dva 4744 | . 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
16 | 3, 15 | eqtrd 2656 | 1 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supp csupp 7295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-supp 7296 |
This theorem is referenced by: gsumzres 18310 gsumzcl2 18311 gsumzf1o 18313 gsumzaddlem 18321 gsumzmhm 18337 gsumzoppg 18344 |
Copyright terms: Public domain | W3C validator |