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Mathbox for Norm Megill |
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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 38864 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
2 | 1 | adantr 479 | . . . 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 39270 | . . . 4 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅) |
6 | 2, 5 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘ 0 ) = ∅) |
7 | 6 | eqeq2d 2739 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = (𝑀‘ 0 ) ↔ (𝑀‘𝑋) = ∅)) |
8 | hlop 38866 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 479 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
10 | pmapeq0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
11 | 10, 3 | op0cl 38688 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
13 | 10, 4 | pmap11 39267 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝑀‘𝑋) = (𝑀‘ 0 ) ↔ 𝑋 = 0 )) |
14 | 12, 13 | mpd3an3 1458 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = (𝑀‘ 0 ) ↔ 𝑋 = 0 )) |
15 | 7, 14 | bitr3d 280 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = ∅ ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∅c0 4326 ‘cfv 6553 Basecbs 17187 0.cp0 18422 OPcops 38676 AtLatcal 38768 HLchlt 38854 pmapcpmap 39002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-pmap 39009 |
This theorem is referenced by: pmapjat1 39358 |
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