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| Mirrors > Home > MPE Home > Th. List > lssats2 | Structured version Visualization version GIF version | ||
| Description: A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.) |
| Ref | Expression |
|---|---|
| lssats2.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
| lssats2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
| lssats2.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
| lssats2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
| Ref | Expression |
|---|---|
| lssats2 | ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈) | |
| 2 | lssats2.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 3 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑊 ∈ LMod) |
| 4 | lssats2.u | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
| 5 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 6 | lssats2.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 7 | 5, 6 | lssel 20935 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
| 8 | 4, 7 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
| 9 | lssats2.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 10 | 5, 9 | lspsnid 20991 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (𝑁‘{𝑦})) |
| 11 | 3, 8, 10 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (𝑁‘{𝑦})) |
| 12 | sneq 4636 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
| 13 | 12 | fveq2d 6910 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{𝑥}) = (𝑁‘{𝑦})) |
| 14 | 13 | eleq2d 2827 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ (𝑁‘{𝑥}) ↔ 𝑦 ∈ (𝑁‘{𝑦}))) |
| 15 | 14 | rspcev 3622 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑈 ∧ 𝑦 ∈ (𝑁‘{𝑦})) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
| 16 | 1, 11, 15 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
| 17 | 16 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑈 → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
| 18 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑊 ∈ LMod) |
| 19 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
| 20 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
| 21 | 6, 9, 18, 19, 20 | ellspsn5 20994 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑁‘{𝑥}) ⊆ 𝑈) |
| 22 | 21 | sseld 3982 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
| 23 | 22 | rexlimdva 3155 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
| 24 | 17, 23 | impbid 212 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
| 25 | eliun 4995 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) | |
| 26 | 24, 25 | bitr4di 289 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}))) |
| 27 | 26 | eqrdv 2735 | 1 ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {csn 4626 ∪ ciun 4991 ‘cfv 6561 Basecbs 17247 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 df-lss 20930 df-lsp 20970 |
| This theorem is referenced by: (None) |
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