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Mirrors > Home > ILE Home > Th. List > lspun0 | GIF version |
Description: The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.) |
Ref | Expression |
---|---|
lspun0.v | ⊢ 𝑉 = (Base‘𝑊) |
lspun0.o | ⊢ 0 = (0g‘𝑊) |
lspun0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspun0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspun0.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
Ref | Expression |
---|---|
lspun0 | ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspun0.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lspun0.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
3 | lspun0.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lspun0.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
5 | 3, 4 | lmod0vcl 13593 | . . . . 5 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
6 | 1, 5 | syl 14 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑉) |
7 | 6 | snssd 3751 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑉) |
8 | lspun0.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 3, 8 | lspun 13678 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉 ∧ { 0 } ⊆ 𝑉) → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 })))) |
10 | 1, 2, 7, 9 | syl3anc 1248 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 })))) |
11 | 4, 8 | lspsn0 13698 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
12 | 1, 11 | syl 14 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
13 | 12 | uneq2d 3303 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ (𝑁‘{ 0 })) = ((𝑁‘𝑋) ∪ { 0 })) |
14 | eqid 2188 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 3, 14, 8 | lspcl 13667 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) |
16 | 1, 2, 15 | syl2anc 411 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) |
17 | 4, 14 | lss0ss 13647 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘𝑋)) |
18 | 1, 16, 17 | syl2anc 411 | . . . . . 6 ⊢ (𝜑 → { 0 } ⊆ (𝑁‘𝑋)) |
19 | ssequn2 3322 | . . . . . 6 ⊢ ({ 0 } ⊆ (𝑁‘𝑋) ↔ ((𝑁‘𝑋) ∪ { 0 }) = (𝑁‘𝑋)) | |
20 | 18, 19 | sylib 122 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ { 0 }) = (𝑁‘𝑋)) |
21 | 13, 20 | eqtrd 2221 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ (𝑁‘{ 0 })) = (𝑁‘𝑋)) |
22 | 21 | fveq2d 5533 | . . 3 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 }))) = (𝑁‘(𝑁‘𝑋))) |
23 | 3, 8 | lspidm 13677 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘𝑋)) |
24 | 1, 2, 23 | syl2anc 411 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = (𝑁‘𝑋)) |
25 | 22, 24 | eqtrd 2221 | . 2 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 }))) = (𝑁‘𝑋)) |
26 | 10, 25 | eqtrd 2221 | 1 ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2159 ∪ cun 3141 ⊆ wss 3143 {csn 3606 ‘cfv 5230 Basecbs 12479 0gc0g 12726 LModclmod 13563 LSubSpclss 13628 LSpanclspn 13662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-i2m1 7933 ax-0lt1 7934 ax-0id 7936 ax-rnegex 7937 ax-pre-ltirr 7940 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-pnf 8011 df-mnf 8012 df-ltxr 8014 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-5 8998 df-6 8999 df-ndx 12482 df-slot 12483 df-base 12485 df-sets 12486 df-plusg 12567 df-mulr 12568 df-sca 12570 df-vsca 12571 df-0g 12728 df-mgm 12797 df-sgrp 12830 df-mnd 12843 df-grp 12913 df-minusg 12914 df-sbg 12915 df-mgp 13235 df-ur 13274 df-ring 13312 df-lmod 13565 df-lssm 13629 df-lsp 13663 |
This theorem is referenced by: (None) |
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