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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 20121 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐴 · 𝑌) ∈ 𝑉) |
10 | 1, 3, 4, 9 | syl3anc 1369 | . . 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 2739 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 20159 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝐴 · 𝑌) ∈ 𝑉) → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
16 | 1, 2, 10, 15 | syl3anc 1369 | . 2 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
17 | 6 | lmodring 20112 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
18 | 1, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Ring) |
19 | ringgrp 19769 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ Grp) |
21 | 8, 14 | ringidcl 19788 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
22 | 18, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
23 | 8, 13 | grpinvcl 18608 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
24 | 20, 22, 23 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
25 | eqid 2739 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 20129 | . . . . 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 1370 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 19814 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) = (𝑁‘𝐴)) |
29 | 28 | oveq1d 7283 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘𝐴) · 𝑌)) |
30 | 27, 29 | eqtr3d 2781 | . . 3 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)) = ((𝑁‘𝐴) · 𝑌)) |
31 | 30 | oveq2d 7284 | . 2 ⊢ (𝜑 → (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
32 | 16, 31 | eqtrd 2779 | 1 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 +gcplusg 16943 .rcmulr 16944 Scalarcsca 16946 ·𝑠 cvsca 16947 Grpcgrp 18558 invgcminusg 18559 -gcsg 18560 1rcur 19718 Ringcrg 19764 LModclmod 20104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-sbg 18563 df-mgp 19702 df-ur 19719 df-ring 19766 df-lmod 20106 |
This theorem is referenced by: lspexch 20372 baerlem5alem1 39701 baerlem5blem1 39702 |
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