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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islshp | Structured version Visualization version GIF version |
Description: The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.) |
Ref | Expression |
---|---|
lshpset.v | ⊢ 𝑉 = (Base‘𝑊) |
lshpset.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lshpset.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lshpset.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
Ref | Expression |
---|---|
islshp | ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpset.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lshpset.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lshpset.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | lshpset.h | . . . 4 ⊢ 𝐻 = (LSHyp‘𝑊) | |
5 | 1, 2, 3, 4 | lshpset 34583 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}) |
6 | 5 | eleq2d 2716 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ 𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})) |
7 | neeq1 2885 | . . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ≠ 𝑉 ↔ 𝑈 ≠ 𝑉)) | |
8 | uneq1 3793 | . . . . . . . 8 ⊢ (𝑠 = 𝑈 → (𝑠 ∪ {𝑣}) = (𝑈 ∪ {𝑣})) | |
9 | 8 | fveq2d 6233 | . . . . . . 7 ⊢ (𝑠 = 𝑈 → (𝑁‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑈 ∪ {𝑣}))) |
10 | 9 | eqeq1d 2653 | . . . . . 6 ⊢ (𝑠 = 𝑈 → ((𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)) |
11 | 10 | rexbidv 3081 | . . . . 5 ⊢ (𝑠 = 𝑈 → (∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)) |
12 | 7, 11 | anbi12d 747 | . . . 4 ⊢ (𝑠 = 𝑈 → ((𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉) ↔ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
13 | 12 | elrab 3396 | . . 3 ⊢ (𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈 ∈ 𝑆 ∧ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
14 | 3anass 1059 | . . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) ↔ (𝑈 ∈ 𝑆 ∧ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) | |
15 | 13, 14 | bitr4i 267 | . 2 ⊢ (𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)) |
16 | 6, 15 | syl6bb 276 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∃wrex 2942 {crab 2945 ∪ cun 3605 {csn 4210 ‘cfv 5926 Basecbs 15904 LSubSpclss 18980 LSpanclspn 19019 LSHypclsh 34580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-lshyp 34582 |
This theorem is referenced by: islshpsm 34585 lshplss 34586 lshpne 34587 lshpnel2N 34590 lkrshp 34710 lshpset2N 34724 dochsatshp 37057 |
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