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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 20773 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐴 · 𝑌) ∈ 𝑉) |
10 | 1, 3, 4, 9 | syl3anc 1368 | . . 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 2725 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 20812 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝐴 · 𝑌) ∈ 𝑉) → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
16 | 1, 2, 10, 15 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
17 | 6 | lmodring 20763 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
18 | 1, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Ring) |
19 | ringgrp 20190 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ Grp) |
21 | 8, 14 | ringidcl 20214 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
22 | 18, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
23 | 8, 13 | grpinvcl 18952 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
24 | 20, 22, 23 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
25 | eqid 2725 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 20782 | . . . . 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 1369 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 20250 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) = (𝑁‘𝐴)) |
29 | 28 | oveq1d 7434 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘𝐴) · 𝑌)) |
30 | 27, 29 | eqtr3d 2767 | . . 3 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)) = ((𝑁‘𝐴) · 𝑌)) |
31 | 30 | oveq2d 7435 | . 2 ⊢ (𝜑 → (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
32 | 16, 31 | eqtrd 2765 | 1 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 +gcplusg 17236 .rcmulr 17237 Scalarcsca 17239 ·𝑠 cvsca 17240 Grpcgrp 18898 invgcminusg 18899 -gcsg 18900 1rcur 20133 Ringcrg 20185 LModclmod 20755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-lmod 20757 |
This theorem is referenced by: lspexch 21029 baerlem5alem1 41311 baerlem5blem1 41312 |
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