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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincresunit1 | Structured version Visualization version GIF version |
Description: Property 1 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 |
---|---|
lincresunit1 | ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lincresunit.g | . 2 ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) | |
2 | eldifi 4106 | . . . . 5 ⊢ (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝑆) | |
3 | lincresunit.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
4 | lincresunit.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑀) | |
5 | lincresunit.e | . . . . . 6 ⊢ 𝐸 = (Base‘𝑅) | |
6 | lincresunit.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | lincresunit.0 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
8 | lincresunit.z | . . . . . 6 ⊢ 𝑍 = (0g‘𝑀) | |
9 | lincresunit.n | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
10 | lincresunit.i | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
11 | lincresunit.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 1 | lincresunitlem2 44538 | . . . . 5 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑠 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ 𝐸) |
13 | 2, 12 | sylan2 594 | . . . 4 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ 𝐸) |
14 | 13 | fmpttd 6882 | . . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))):(𝑆 ∖ {𝑋})⟶𝐸) |
15 | 5 | fvexi 6687 | . . . 4 ⊢ 𝐸 ∈ V |
16 | difexg 5234 | . . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V) | |
17 | 16 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V) |
18 | 17 | adantr 483 | . . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝑆 ∖ {𝑋}) ∈ V) |
19 | elmapg 8422 | . . . 4 ⊢ ((𝐸 ∈ V ∧ (𝑆 ∖ {𝑋}) ∈ V) → ((𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})) ↔ (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))):(𝑆 ∖ {𝑋})⟶𝐸)) | |
20 | 15, 18, 19 | sylancr 589 | . . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → ((𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})) ↔ (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))):(𝑆 ∖ {𝑋})⟶𝐸)) |
21 | 14, 20 | mpbird 259 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) |
22 | 1, 21 | eqeltrid 2920 | 1 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∖ cdif 3936 𝒫 cpw 4542 {csn 4570 ↦ cmpt 5149 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 Basecbs 16486 .rcmulr 16569 Scalarcsca 16571 0gc0g 16716 invgcminusg 18107 Unitcui 19392 invrcinvr 19424 LModclmod 19637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-lmod 19639 |
This theorem is referenced by: lincresunit3lem2 44542 lincresunit3 44543 isldepslvec2 44547 |
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