| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > dprdff | Structured version Visualization version GIF version | ||
| Description: A finitely supported function in 𝑆 is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.) |
| Ref | Expression |
|---|---|
| dprdff.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
| dprdff.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
| dprdff.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
| dprdff.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
| dprdff.b | ⊢ 𝐵 = (Base‘𝐺) |
| Ref | Expression |
|---|---|
| dprdff | ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dprdff.3 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
| 2 | dprdff.w | . . . . 5 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
| 3 | dprdff.1 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
| 4 | dprdff.2 | . . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
| 5 | 2, 3, 4 | dprdw 20030 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 ))) |
| 6 | 1, 5 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )) |
| 7 | 6 | simp1d 1143 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
| 8 | 6 | simp2d 1144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) |
| 9 | 3, 4 | dprdf2 20027 | . . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
| 10 | 9 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
| 11 | dprdff.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
| 12 | 11 | subgss 19145 | . . . . . 6 ⊢ ((𝑆‘𝑥) ∈ (SubGrp‘𝐺) → (𝑆‘𝑥) ⊆ 𝐵) |
| 13 | 10, 12 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ 𝐵) |
| 14 | 13 | sseld 3982 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) ∈ (𝑆‘𝑥) → (𝐹‘𝑥) ∈ 𝐵)) |
| 15 | 14 | ralimdva 3167 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐵)) |
| 16 | 8, 15 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐵) |
| 17 | ffnfv 7139 | . 2 ⊢ (𝐹:𝐼⟶𝐵 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐵)) | |
| 18 | 7, 16, 17 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Xcixp 8937 finSupp cfsupp 9401 Basecbs 17247 SubGrpcsubg 19138 DProd cdprd 20013 |
| 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-pow 5365 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-nel 3047 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-1st 8014 df-2nd 8015 df-ixp 8938 df-subg 19141 df-dprd 20015 |
| This theorem is referenced by: dprdfcntz 20035 dprdssv 20036 dprdfid 20037 dprdfinv 20039 dprdfadd 20040 dprdfsub 20041 dprdfeq0 20042 dprdf11 20043 dprdlub 20046 dmdprdsplitlem 20057 dprddisj2 20059 dpjidcl 20078 |
| Copyright terms: Public domain | W3C validator |