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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 37103 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
9 | 1, 2, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
10 | ellkr2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | biantrurd 533 | . 2 ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
12 | 9, 11 | bitr4d 281 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 Basecbs 16912 Scalarcsca 16965 0gc0g 17150 LFnlclfn 37071 LKerclk 37099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-lfl 37072 df-lkr 37100 |
This theorem is referenced by: lclkrlem2f 39526 lclkrlem2n 39534 lcfrlem3 39558 lcfrlem25 39581 hdmapellkr 39928 hdmapip0 39929 hdmapinvlem1 39932 |
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