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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincresunitlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
Ref | Expression |
---|---|
lincresunit.b | ⊢ 𝐵 = (Base‘𝑀) |
lincresunit.r | ⊢ 𝑅 = (Scalar‘𝑀) |
lincresunit.e | ⊢ 𝐸 = (Base‘𝑅) |
lincresunit.u | ⊢ 𝑈 = (Unit‘𝑅) |
lincresunit.0 | ⊢ 0 = (0g‘𝑅) |
lincresunit.z | ⊢ 𝑍 = (0g‘𝑀) |
lincresunit.n | ⊢ 𝑁 = (invg‘𝑅) |
lincresunit.i | ⊢ 𝐼 = (invr‘𝑅) |
lincresunit.t | ⊢ · = (.r‘𝑅) |
lincresunit.g | ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) |
Ref | Expression |
---|---|
lincresunitlem1 | ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lincresunit.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑀) | |
2 | 1 | lmodring 19912 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring) |
3 | 2 | 3ad2ant2 1136 | . . 3 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Ring) |
4 | 3 | adantr 484 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝑅 ∈ Ring) |
5 | simpr 488 | . . 3 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈) → (𝐹‘𝑋) ∈ 𝑈) | |
6 | lincresunit.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | lincresunit.n | . . . 4 ⊢ 𝑁 = (invg‘𝑅) | |
8 | 6, 7 | unitnegcl 19704 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐹‘𝑋) ∈ 𝑈) → (𝑁‘(𝐹‘𝑋)) ∈ 𝑈) |
9 | 3, 5, 8 | syl2an 599 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝑁‘(𝐹‘𝑋)) ∈ 𝑈) |
10 | lincresunit.i | . . 3 ⊢ 𝐼 = (invr‘𝑅) | |
11 | lincresunit.e | . . 3 ⊢ 𝐸 = (Base‘𝑅) | |
12 | 6, 10, 11 | ringinvcl 19699 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘(𝐹‘𝑋)) ∈ 𝑈) → (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸) |
13 | 4, 9, 12 | syl2anc 587 | 1 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∖ cdif 3868 𝒫 cpw 4518 {csn 4546 ↦ cmpt 5140 ‘cfv 6385 (class class class)co 7218 ↑m cmap 8513 Basecbs 16765 .rcmulr 16808 Scalarcsca 16810 0gc0g 16949 invgcminusg 18371 Ringcrg 19567 Unitcui 19662 invrcinvr 19694 LModclmod 19904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-om 7650 df-tpos 7973 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-er 8396 df-en 8632 df-dom 8633 df-sdom 8634 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-nn 11836 df-2 11898 df-3 11899 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-0g 16951 df-mgm 18119 df-sgrp 18168 df-mnd 18179 df-grp 18373 df-minusg 18374 df-mgp 19510 df-ur 19522 df-ring 19569 df-oppr 19646 df-dvdsr 19664 df-unit 19665 df-invr 19695 df-lmod 19906 |
This theorem is referenced by: lincresunitlem2 45498 lincresunit2 45500 |
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