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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincresunitlem2 | Structured version Visualization version GIF version |
Description: Lemma 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 |
---|---|
lincresunitlem2 | ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑌 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑌)) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lincresunit.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑀) | |
2 | 1 | lmodring 19636 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring) |
3 | 2 | 3ad2ant2 1130 | . . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Ring) |
4 | 3 | adantr 483 | . . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝑅 ∈ Ring) |
5 | 4 | adantr 483 | . 2 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑌 ∈ 𝑆) → 𝑅 ∈ Ring) |
6 | lincresunit.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
7 | lincresunit.e | . . . 4 ⊢ 𝐸 = (Base‘𝑅) | |
8 | lincresunit.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
9 | lincresunit.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
10 | lincresunit.z | . . . 4 ⊢ 𝑍 = (0g‘𝑀) | |
11 | lincresunit.n | . . . 4 ⊢ 𝑁 = (invg‘𝑅) | |
12 | lincresunit.i | . . . 4 ⊢ 𝐼 = (invr‘𝑅) | |
13 | lincresunit.t | . . . 4 ⊢ · = (.r‘𝑅) | |
14 | lincresunit.g | . . . 4 ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) | |
15 | 6, 1, 7, 8, 9, 10, 11, 12, 13, 14 | lincresunitlem1 44523 | . . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸) |
16 | 15 | adantr 483 | . 2 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑌 ∈ 𝑆) → (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸) |
17 | elmapi 8422 | . . . . 5 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → 𝐹:𝑆⟶𝐸) | |
18 | ffvelrn 6844 | . . . . . 6 ⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑌 ∈ 𝑆) → (𝐹‘𝑌) ∈ 𝐸) | |
19 | 18 | ex 415 | . . . . 5 ⊢ (𝐹:𝑆⟶𝐸 → (𝑌 ∈ 𝑆 → (𝐹‘𝑌) ∈ 𝐸)) |
20 | 17, 19 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → (𝑌 ∈ 𝑆 → (𝐹‘𝑌) ∈ 𝐸)) |
21 | 20 | ad2antrl 726 | . . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝑌 ∈ 𝑆 → (𝐹‘𝑌) ∈ 𝐸)) |
22 | 21 | imp 409 | . 2 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑌 ∈ 𝑆) → (𝐹‘𝑌) ∈ 𝐸) |
23 | 7, 13 | ringcl 19305 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸 ∧ (𝐹‘𝑌) ∈ 𝐸) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑌)) ∈ 𝐸) |
24 | 5, 16, 22, 23 | syl3anc 1367 | 1 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑌 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑌)) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 𝒫 cpw 4539 {csn 4561 ↦ cmpt 5139 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 Basecbs 16477 .rcmulr 16560 Scalarcsca 16562 0gc0g 16707 invgcminusg 18098 Ringcrg 19291 Unitcui 19383 invrcinvr 19415 LModclmod 19628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-lmod 19630 |
This theorem is referenced by: lincresunit1 44525 lincresunit2 44526 lincresunit3lem1 44527 lincresunit3 44529 |
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