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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapeq0 | Structured version Visualization version GIF version |
Description: A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.) |
Ref | Expression |
---|---|
pmapeq0.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapeq0.z | ⊢ 0 = (0.‘𝐾) |
pmapeq0.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmapeq0 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = ∅ ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlatl 33665 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
2 | 1 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ AtLat) |
3 | pmapeq0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | pmapeq0.m | . . . . 5 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | 3, 4 | pmap0 34069 | . . . 4 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅) |
6 | 2, 5 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘ 0 ) = ∅) |
7 | 6 | eqeq2d 2620 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = (𝑀‘ 0 ) ↔ (𝑀‘𝑋) = ∅)) |
8 | hlop 33667 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
10 | pmapeq0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
11 | 10, 3 | op0cl 33489 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
13 | 10, 4 | pmap11 34066 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝑀‘𝑋) = (𝑀‘ 0 ) ↔ 𝑋 = 0 )) |
14 | 12, 13 | mpd3an3 1417 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = (𝑀‘ 0 ) ↔ 𝑋 = 0 )) |
15 | 7, 14 | bitr3d 269 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = ∅ ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ‘cfv 5804 Basecbs 15695 0.cp0 16860 OPcops 33477 AtLatcal 33569 HLchlt 33655 pmapcpmap 33801 |
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-riota 6511 df-ov 6552 df-oprab 6553 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-pmap 33808 |
This theorem is referenced by: pmapjat1 34157 |
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