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Mirrors > Home > MPE Home > Th. List > lmodsubvs | Structured version Visualization version GIF version |
Description: Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.) |
Ref | Expression |
---|---|
lmodsubvs.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodsubvs.p | ⊢ + = (+g‘𝑊) |
lmodsubvs.m | ⊢ − = (-g‘𝑊) |
lmodsubvs.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodsubvs.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodsubvs.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodsubvs.n | ⊢ 𝑁 = (invg‘𝐹) |
lmodsubvs.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodsubvs.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lmodsubvs.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lmodsubvs.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lmodsubvs | ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodsubvs.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodsubvs.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | lmodsubvs.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
4 | lmodsubvs.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | lmodsubvs.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
6 | lmodsubvs.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | lmodsubvs.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
8 | lmodsubvs.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
9 | 5, 6, 7, 8 | lmodvscl 20140 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐴 · 𝑌) ∈ 𝑉) |
10 | 1, 3, 4, 9 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑌) ∈ 𝑉) |
11 | lmodsubvs.p | . . . 4 ⊢ + = (+g‘𝑊) | |
12 | lmodsubvs.m | . . . 4 ⊢ − = (-g‘𝑊) | |
13 | lmodsubvs.n | . . . 4 ⊢ 𝑁 = (invg‘𝐹) | |
14 | eqid 2738 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 20178 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝐴 · 𝑌) ∈ 𝑉) → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
16 | 1, 2, 10, 15 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
17 | 6 | lmodring 20131 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
18 | 1, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Ring) |
19 | ringgrp 19788 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ Grp) |
21 | 8, 14 | ringidcl 19807 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
22 | 18, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
23 | 8, 13 | grpinvcl 18627 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
24 | 20, 22, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
25 | eqid 2738 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 20148 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ ((𝑁‘(1r‘𝐹)) ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉)) → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) |
27 | 1, 24, 3, 4, 26 | syl13anc 1371 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 19833 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) = (𝑁‘𝐴)) |
29 | 28 | oveq1d 7290 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘𝐴) · 𝑌)) |
30 | 27, 29 | eqtr3d 2780 | . . 3 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)) = ((𝑁‘𝐴) · 𝑌)) |
31 | 30 | oveq2d 7291 | . 2 ⊢ (𝜑 → (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
32 | 16, 31 | eqtrd 2778 | 1 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 .rcmulr 16963 Scalarcsca 16965 ·𝑠 cvsca 16966 Grpcgrp 18577 invgcminusg 18578 -gcsg 18579 1rcur 19737 Ringcrg 19783 LModclmod 20123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mgp 19721 df-ur 19738 df-ring 19785 df-lmod 20125 |
This theorem is referenced by: lspexch 20391 baerlem5alem1 39722 baerlem5blem1 39723 |
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