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Mirrors > Home > MPE Home > Th. List > Mathboxes > ellkr2 | Structured version Visualization version GIF version |
Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.) |
Ref | Expression |
---|---|
lkrfval2.v | ⊢ 𝑉 = (Base‘𝑊) |
lkrfval2.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkrfval2.o | ⊢ 0 = (0g‘𝐷) |
lkrfval2.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrfval2.k | ⊢ 𝐾 = (LKer‘𝑊) |
ellkr2.w | ⊢ (𝜑 → 𝑊 ∈ 𝑌) |
ellkr2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ellkr2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
ellkr2 | ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellkr2.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑌) | |
2 | ellkr2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
3 | lkrfval2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lkrfval2.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
5 | lkrfval2.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
6 | lkrfval2.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
7 | lkrfval2.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
8 | 3, 4, 5, 6, 7 | ellkr 33394 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
9 | 1, 2, 8 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
10 | ellkr2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | biantrurd 528 | . 2 ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
12 | 9, 11 | bitr4d 270 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 Scalarcsca 15771 0gc0g 15923 LFnlclfn 33362 LKerclk 33390 |
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-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-pw 4110 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-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-lfl 33363 df-lkr 33391 |
This theorem is referenced by: lclkrlem2f 35819 lclkrlem2n 35827 lcfrlem3 35851 lcfrlem25 35874 hdmapellkr 36224 hdmapip0 36225 hdmapinvlem1 36228 |
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