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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 20045 | . . . 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 1141 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
8 | 6 | simp2d 1142 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) |
9 | 3, 4 | dprdf2 20042 | . . . . . . 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 19158 | . . . . . 6 ⊢ ((𝑆‘𝑥) ∈ (SubGrp‘𝐺) → (𝑆‘𝑥) ⊆ 𝐵) |
13 | 10, 12 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ 𝐵) |
14 | 13 | sseld 3994 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) ∈ (𝑆‘𝑥) → (𝐹‘𝑥) ∈ 𝐵)) |
15 | 14 | ralimdva 3165 | . . 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 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 class class class wbr 5148 dom cdm 5689 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 Xcixp 8936 finSupp cfsupp 9399 Basecbs 17245 SubGrpcsubg 19151 DProd cdprd 20028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-ixp 8937 df-subg 19154 df-dprd 20030 |
This theorem is referenced by: dprdfcntz 20050 dprdssv 20051 dprdfid 20052 dprdfinv 20054 dprdfadd 20055 dprdfsub 20056 dprdfeq0 20057 dprdf11 20058 dprdlub 20061 dmdprdsplitlem 20072 dprddisj2 20074 dpjidcl 20093 |
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