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Mirrors > Home > MPE Home > Th. List > clmvsubval | Structured version Visualization version GIF version |
Description: Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 19691. (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.) |
Ref | Expression |
---|---|
clmvsubval.v | ⊢ 𝑉 = (Base‘𝑊) |
clmvsubval.p | ⊢ + = (+g‘𝑊) |
clmvsubval.m | ⊢ − = (-g‘𝑊) |
clmvsubval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
clmvsubval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
clmvsubval | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 23673 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | clmvsubval.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | clmvsubval.p | . . . 4 ⊢ + = (+g‘𝑊) | |
4 | clmvsubval.m | . . . 4 ⊢ − = (-g‘𝑊) | |
5 | clmvsubval.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | clmvsubval.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
7 | eqid 2823 | . . . 4 ⊢ (invg‘𝐹) = (invg‘𝐹) | |
8 | eqid 2823 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 2, 3, 4, 5, 6, 7, 8 | lmodvsubval2 19691 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵))) |
10 | 1, 9 | syl3an1 1159 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵))) |
11 | 5 | clm1 23679 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
12 | 11 | eqcomd 2829 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → (1r‘𝐹) = 1) |
13 | 12 | fveq2d 6676 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → ((invg‘𝐹)‘(1r‘𝐹)) = ((invg‘𝐹)‘1)) |
14 | 5 | clmring 23676 | . . . . . . . . 9 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring) |
15 | eqid 2823 | . . . . . . . . . 10 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
16 | 15, 8 | ringidcl 19320 | . . . . . . . . 9 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ (Base‘𝐹)) |
17 | 14, 16 | syl 17 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂMod → (1r‘𝐹) ∈ (Base‘𝐹)) |
18 | 11, 17 | eqeltrd 2915 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 ∈ (Base‘𝐹)) |
19 | 5, 15 | clmneg 23687 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂMod ∧ 1 ∈ (Base‘𝐹)) → -1 = ((invg‘𝐹)‘1)) |
20 | 18, 19 | mpdan 685 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → -1 = ((invg‘𝐹)‘1)) |
21 | 13, 20 | eqtr4d 2861 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → ((invg‘𝐹)‘(1r‘𝐹)) = -1) |
22 | 21 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((invg‘𝐹)‘(1r‘𝐹)) = -1) |
23 | 22 | oveq1d 7173 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵) = (-1 · 𝐵)) |
24 | 23 | oveq2d 7174 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵)) = (𝐴 + (-1 · 𝐵))) |
25 | 10, 24 | eqtrd 2858 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 1c1 10540 -cneg 10873 Basecbs 16485 +gcplusg 16567 Scalarcsca 16570 ·𝑠 cvsca 16571 invgcminusg 18106 -gcsg 18107 1rcur 19253 Ringcrg 19299 LModclmod 19636 ℂModcclm 23668 |
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 ax-addf 10618 ax-mulf 10619 |
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-int 4879 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-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 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-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-lmod 19638 df-cnfld 20548 df-clm 23669 |
This theorem is referenced by: clmvsubval2 23716 ncvsdif 23761 ncvspds 23767 cphipval 23848 |
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