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Mirrors > Home > MPE Home > Th. List > lmodnegadd | Structured version Visualization version GIF version |
Description: Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.) |
Ref | Expression |
---|---|
lmodnegadd.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodnegadd.p | ⊢ + = (+g‘𝑊) |
lmodnegadd.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodnegadd.n | ⊢ 𝑁 = (invg‘𝑊) |
lmodnegadd.r | ⊢ 𝑅 = (Scalar‘𝑊) |
lmodnegadd.k | ⊢ 𝐾 = (Base‘𝑅) |
lmodnegadd.i | ⊢ 𝐼 = (invg‘𝑅) |
lmodnegadd.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodnegadd.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lmodnegadd.b | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
lmodnegadd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lmodnegadd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lmodnegadd | ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodnegadd.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodabl 19610 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑊 ∈ Abel) |
4 | lmodnegadd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
5 | lmodnegadd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
6 | lmodnegadd.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
7 | lmodnegadd.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑊) | |
8 | lmodnegadd.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | lmodnegadd.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
10 | 6, 7, 8, 9 | lmodvscl 19580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
11 | 1, 4, 5, 10 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
12 | lmodnegadd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
13 | lmodnegadd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | 6, 7, 8, 9 | lmodvscl 19580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐵 · 𝑌) ∈ 𝑉) |
15 | 1, 12, 13, 14 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ 𝑉) |
16 | lmodnegadd.p | . . . 4 ⊢ + = (+g‘𝑊) | |
17 | lmodnegadd.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
18 | 6, 16, 17 | ablinvadd 18859 | . . 3 ⊢ ((𝑊 ∈ Abel ∧ (𝐴 · 𝑋) ∈ 𝑉 ∧ (𝐵 · 𝑌) ∈ 𝑉) → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = ((𝑁‘(𝐴 · 𝑋)) + (𝑁‘(𝐵 · 𝑌)))) |
19 | 3, 11, 15, 18 | syl3anc 1363 | . 2 ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = ((𝑁‘(𝐴 · 𝑋)) + (𝑁‘(𝐵 · 𝑌)))) |
20 | lmodnegadd.i | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
21 | 6, 7, 8, 17, 9, 20, 1, 5, 4 | lmodvsneg 19607 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐴 · 𝑋)) = ((𝐼‘𝐴) · 𝑋)) |
22 | 6, 7, 8, 17, 9, 20, 1, 13, 12 | lmodvsneg 19607 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐵 · 𝑌)) = ((𝐼‘𝐵) · 𝑌)) |
23 | 21, 22 | oveq12d 7163 | . 2 ⊢ (𝜑 → ((𝑁‘(𝐴 · 𝑋)) + (𝑁‘(𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌))) |
24 | 19, 23 | eqtrd 2853 | 1 ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 ·𝑠 cvsca 16557 invgcminusg 18042 Abelcabl 18836 LModclmod 19563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 |
This theorem is referenced by: baerlem3lem1 38723 |
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