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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 19653 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐴 · 𝑌) ∈ 𝑉) |
10 | 1, 3, 4, 9 | syl3anc 1367 | . . 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 2823 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 19691 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝐴 · 𝑌) ∈ 𝑉) → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
16 | 1, 2, 10, 15 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
17 | 6 | lmodring 19644 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
18 | 1, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Ring) |
19 | ringgrp 19304 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ Grp) |
21 | 8, 14 | ringidcl 19320 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
22 | 18, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
23 | 8, 13 | grpinvcl 18153 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
24 | 20, 22, 23 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
25 | eqid 2823 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 19661 | . . . . 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 1368 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 19346 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) = (𝑁‘𝐴)) |
29 | 28 | oveq1d 7173 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘𝐴) · 𝑌)) |
30 | 27, 29 | eqtr3d 2860 | . . 3 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)) = ((𝑁‘𝐴) · 𝑌)) |
31 | 30 | oveq2d 7174 | . 2 ⊢ (𝜑 → (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
32 | 16, 31 | eqtrd 2858 | 1 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 Grpcgrp 18105 invgcminusg 18106 -gcsg 18107 1rcur 19253 Ringcrg 19299 LModclmod 19636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mgp 19242 df-ur 19254 df-ring 19301 df-lmod 19638 |
This theorem is referenced by: lspexch 19903 baerlem5alem1 38846 baerlem5blem1 38847 |
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