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Mirrors > Home > MPE Home > Th. List > dprdwd | Structured version Visualization version GIF version |
Description: A mapping being a finitely supported function in the family 𝑆. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.) (Proof shortened by OpenAI, 30-Mar-2020.) |
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) |
dprdwd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝑆‘𝑥)) |
dprdwd.4 | ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 ) |
Ref | Expression |
---|---|
dprdwd | ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2611 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) = (𝑥 ∈ 𝐼 ↦ 𝐴)) | |
2 | dprdwd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝑆‘𝑥)) | |
3 | 2 | ralrimiva 2949 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥)) |
4 | dprdff.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
5 | dprdff.2 | . . . . . . . 8 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
6 | 4, 5 | dprddomcld 18223 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V) |
7 | mptelixpg 7831 | . . . . . . 7 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥))) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥))) |
9 | 3, 8 | mpbird 246 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥)) |
10 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑆‘𝑥) = (𝑆‘𝑖)) | |
11 | 10 | cbvixpv 7812 | . . . . 5 ⊢ X𝑥 ∈ 𝐼 (𝑆‘𝑥) = X𝑖 ∈ 𝐼 (𝑆‘𝑖) |
12 | 9, 11 | syl6eleq 2698 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖)) |
13 | dprdwd.4 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 ) | |
14 | breq1 4586 | . . . . 5 ⊢ (ℎ = (𝑥 ∈ 𝐼 ↦ 𝐴) → (ℎ finSupp 0 ↔ (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 )) | |
15 | 14 | elrab 3331 | . . . 4 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ↔ ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 )) |
16 | 12, 13, 15 | sylanbrc 695 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }) |
17 | dprdff.w | . . 3 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
18 | 16, 17 | syl6eleqr 2699 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊) |
19 | 1, 18 | eqeltrrd 2689 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 Xcixp 7794 finSupp cfsupp 8158 DProd cdprd 18215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-oprab 6553 df-mpt2 6554 df-ixp 7795 df-dprd 18217 |
This theorem is referenced by: dprdfid 18239 dprdfinv 18241 dprdfadd 18242 dmdprdsplitlem 18259 dpjidcl 18280 dchrptlem3 24791 |
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