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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 36726 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
9 | 1, 2, 8 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
10 | ellkr2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | biantrurd 536 | . 2 ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
12 | 9, 11 | bitr4d 285 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 Basecbs 16586 Scalarcsca 16671 0gc0g 16816 LFnlclfn 36694 LKerclk 36722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-map 8439 df-lfl 36695 df-lkr 36723 |
This theorem is referenced by: lclkrlem2f 39149 lclkrlem2n 39157 lcfrlem3 39181 lcfrlem25 39204 hdmapellkr 39551 hdmapip0 39552 hdmapinvlem1 39555 |
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