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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvbr2 | Structured version Visualization version GIF version |
Description: The covers relation for a left vector space (or a left module). (cvbr2 29728 analog.) (Contributed by NM, 9-Jan-2015.) |
Ref | Expression |
---|---|
lcvfbr.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lcvfbr.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lcvfbr.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
lcvfbr.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lcvfbr.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Ref | Expression |
---|---|
lcvbr2 | ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvfbr.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | lcvfbr.c | . . 3 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
3 | lcvfbr.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
4 | lcvfbr.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
5 | lcvfbr.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
6 | 1, 2, 3, 4, 5 | lcvbr 35169 | . 2 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))) |
7 | iman 392 | . . . . . 6 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ¬ ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) ∧ ¬ 𝑠 = 𝑈)) | |
8 | anass 462 | . . . . . . 7 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇 ⊊ 𝑠 ∧ (𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈))) | |
9 | dfpss2 3913 | . . . . . . . 8 ⊢ (𝑠 ⊊ 𝑈 ↔ (𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈)) | |
10 | 9 | anbi2i 616 | . . . . . . 7 ⊢ ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈) ↔ (𝑇 ⊊ 𝑠 ∧ (𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈))) |
11 | 8, 10 | bitr4i 270 | . . . . . 6 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)) |
12 | 7, 11 | xchbinx 326 | . . . . 5 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ¬ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)) |
13 | 12 | ralbii 3161 | . . . 4 ⊢ (∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ∀𝑠 ∈ 𝑆 ¬ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)) |
14 | ralnex 3173 | . . . 4 ⊢ (∀𝑠 ∈ 𝑆 ¬ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)) | |
15 | 13, 14 | bitri 267 | . . 3 ⊢ (∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)) |
16 | 15 | anbi2i 616 | . 2 ⊢ ((𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈)) ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))) |
17 | 6, 16 | syl6bbr 281 | 1 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ∃wrex 3090 ⊆ wss 3791 ⊊ wpss 3792 class class class wbr 4886 ‘cfv 6135 LSubSpclss 19324 ⋖L clcv 35166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-lcv 35167 |
This theorem is referenced by: lsmcv2 35177 lsat0cv 35181 |
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