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Mirrors > Home > MPE Home > Th. List > lspf | Structured version Visualization version GIF version |
Description: The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
Ref | Expression |
---|---|
lspval.v | ⊢ 𝑉 = (Base‘𝑊) |
lspval.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspval.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspf | ⊢ (𝑊 ∈ LMod → 𝑁:𝒫 𝑉⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝒫 𝑉) → 𝑊 ∈ LMod) | |
2 | ssrab2 3650 | . . . . 5 ⊢ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ⊆ 𝑆 | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝒫 𝑉) → {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ⊆ 𝑆) |
4 | lspval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
5 | lspval.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊) | |
6 | 4, 5 | lss1 18760 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
7 | elpwi 4117 | . . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 → 𝑠 ⊆ 𝑉) | |
8 | sseq2 3590 | . . . . . . 7 ⊢ (𝑝 = 𝑉 → (𝑠 ⊆ 𝑝 ↔ 𝑠 ⊆ 𝑉)) | |
9 | 8 | rspcev 3282 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑆 ∧ 𝑠 ⊆ 𝑉) → ∃𝑝 ∈ 𝑆 𝑠 ⊆ 𝑝) |
10 | 6, 7, 9 | syl2an 493 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝒫 𝑉) → ∃𝑝 ∈ 𝑆 𝑠 ⊆ 𝑝) |
11 | rabn0 3912 | . . . . 5 ⊢ ({𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ≠ ∅ ↔ ∃𝑝 ∈ 𝑆 𝑠 ⊆ 𝑝) | |
12 | 10, 11 | sylibr 223 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝒫 𝑉) → {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ≠ ∅) |
13 | 5 | lssintcl 18785 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ⊆ 𝑆 ∧ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ≠ ∅) → ∩ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ∈ 𝑆) |
14 | 1, 3, 12, 13 | syl3anc 1318 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝒫 𝑉) → ∩ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝} ∈ 𝑆) |
15 | eqid 2610 | . . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝}) | |
16 | 14, 15 | fmptd 6292 | . 2 ⊢ (𝑊 ∈ LMod → (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝}):𝒫 𝑉⟶𝑆) |
17 | lspval.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
18 | 4, 5, 17 | lspfval 18794 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝})) |
19 | 18 | feq1d 5943 | . 2 ⊢ (𝑊 ∈ LMod → (𝑁:𝒫 𝑉⟶𝑆 ↔ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑝 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑝}):𝒫 𝑉⟶𝑆)) |
20 | 16, 19 | mpbird 246 | 1 ⊢ (𝑊 ∈ LMod → 𝑁:𝒫 𝑉⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∩ cint 4410 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 Basecbs 15695 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 |
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 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754 df-lsp 18793 |
This theorem is referenced by: lspcl 18797 islmodfg 36657 |
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