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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 18872 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 ))) |
6 | 1, 5 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )) |
7 | 6 | simp1d 1122 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
8 | 6 | simp2d 1123 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) |
9 | 3, 4 | dprdf2 18869 | . . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
10 | 9 | ffvelrnda 6670 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
11 | dprdff.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
12 | 11 | subgss 18054 | . . . . . 6 ⊢ ((𝑆‘𝑥) ∈ (SubGrp‘𝐺) → (𝑆‘𝑥) ⊆ 𝐵) |
13 | 10, 12 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ 𝐵) |
14 | 13 | sseld 3853 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) ∈ (𝑆‘𝑥) → (𝐹‘𝑥) ∈ 𝐵)) |
15 | 14 | ralimdva 3121 | . . 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 6699 | . 2 ⊢ (𝐹:𝐼⟶𝐵 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐵)) | |
18 | 7, 16, 17 | sylanbrc 575 | 1 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ∀wral 3082 {crab 3086 ⊆ wss 3825 class class class wbr 4923 dom cdm 5400 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 Xcixp 8251 finSupp cfsupp 8620 Basecbs 16329 SubGrpcsubg 18047 DProd cdprd 18855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-ixp 8252 df-subg 18050 df-dprd 18857 |
This theorem is referenced by: dprdfcntz 18877 dprdssv 18878 dprdfid 18879 dprdfinv 18881 dprdfadd 18882 dprdfsub 18883 dprdfeq0 18884 dprdf11 18885 dprdlub 18888 dmdprdsplitlem 18899 dprddisj2 18901 dpjidcl 18920 |
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