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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qusvscpbl | Structured version Visualization version GIF version |
Description: The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.) |
Ref | Expression |
---|---|
eqgvscpbl.v | ⊢ 𝐵 = (Base‘𝑀) |
eqgvscpbl.e | ⊢ ∼ = (𝑀 ~QG 𝐺) |
eqgvscpbl.s | ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) |
eqgvscpbl.p | ⊢ · = ( ·𝑠 ‘𝑀) |
eqgvscpbl.m | ⊢ (𝜑 → 𝑀 ∈ LMod) |
eqgvscpbl.g | ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) |
eqgvscpbl.k | ⊢ (𝜑 → 𝐾 ∈ 𝑆) |
qusscaval.n | ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) |
qusscaval.m | ⊢ ∙ = ( ·𝑠 ‘𝑁) |
qusvscpbl.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) |
qusvscpbl.u | ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
qusvscpbl.v | ⊢ (𝜑 → 𝑉 ∈ 𝐵) |
Ref | Expression |
---|---|
qusvscpbl | ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqgvscpbl.v | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
2 | eqid 2798 | . . . 4 ⊢ (𝑀 ~QG 𝐺) = (𝑀 ~QG 𝐺) | |
3 | eqgvscpbl.s | . . . 4 ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) | |
4 | eqgvscpbl.p | . . . 4 ⊢ · = ( ·𝑠 ‘𝑀) | |
5 | eqgvscpbl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
6 | eqgvscpbl.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) | |
7 | eqgvscpbl.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | eqgvscpbl 30970 | . . 3 ⊢ (𝜑 → (𝑈(𝑀 ~QG 𝐺)𝑉 → (𝐾 · 𝑈)(𝑀 ~QG 𝐺)(𝐾 · 𝑉))) |
9 | eqid 2798 | . . . . . . 7 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
10 | 9 | lsssubg 19722 | . . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀)) |
11 | 5, 6, 10 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀)) |
12 | 1, 2 | eqger 18322 | . . . . 5 ⊢ (𝐺 ∈ (SubGrp‘𝑀) → (𝑀 ~QG 𝐺) Er 𝐵) |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 ~QG 𝐺) Er 𝐵) |
14 | qusvscpbl.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵) | |
15 | 13, 14 | erth 8321 | . . 3 ⊢ (𝜑 → (𝑈(𝑀 ~QG 𝐺)𝑉 ↔ [𝑈](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺))) |
16 | eqid 2798 | . . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
17 | 1, 16, 4, 3 | lmodvscl 19644 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑈 ∈ 𝐵) → (𝐾 · 𝑈) ∈ 𝐵) |
18 | 5, 7, 14, 17 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐾 · 𝑈) ∈ 𝐵) |
19 | 13, 18 | erth 8321 | . . 3 ⊢ (𝜑 → ((𝐾 · 𝑈)(𝑀 ~QG 𝐺)(𝐾 · 𝑉) ↔ [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺))) |
20 | 8, 15, 19 | 3imtr3d 296 | . 2 ⊢ (𝜑 → ([𝑈](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺) → [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺))) |
21 | eceq1 8310 | . . . . 5 ⊢ (𝑥 = 𝑈 → [𝑥](𝑀 ~QG 𝐺) = [𝑈](𝑀 ~QG 𝐺)) | |
22 | qusvscpbl.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
23 | ovex 7168 | . . . . . 6 ⊢ (𝑀 ~QG 𝐺) ∈ V | |
24 | ecexg 8276 | . . . . . 6 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [𝑈](𝑀 ~QG 𝐺) ∈ V) | |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ [𝑈](𝑀 ~QG 𝐺) ∈ V |
26 | 21, 22, 25 | fvmpt 6745 | . . . 4 ⊢ (𝑈 ∈ 𝐵 → (𝐹‘𝑈) = [𝑈](𝑀 ~QG 𝐺)) |
27 | 14, 26 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹‘𝑈) = [𝑈](𝑀 ~QG 𝐺)) |
28 | qusvscpbl.v | . . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐵) | |
29 | eceq1 8310 | . . . . 5 ⊢ (𝑥 = 𝑉 → [𝑥](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺)) | |
30 | ecexg 8276 | . . . . . 6 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [𝑉](𝑀 ~QG 𝐺) ∈ V) | |
31 | 23, 30 | ax-mp 5 | . . . . 5 ⊢ [𝑉](𝑀 ~QG 𝐺) ∈ V |
32 | 29, 22, 31 | fvmpt 6745 | . . . 4 ⊢ (𝑉 ∈ 𝐵 → (𝐹‘𝑉) = [𝑉](𝑀 ~QG 𝐺)) |
33 | 28, 32 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹‘𝑉) = [𝑉](𝑀 ~QG 𝐺)) |
34 | 27, 33 | eqeq12d 2814 | . 2 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ [𝑈](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺))) |
35 | eceq1 8310 | . . . . 5 ⊢ (𝑥 = (𝐾 · 𝑈) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑈)](𝑀 ~QG 𝐺)) | |
36 | ecexg 8276 | . . . . . 6 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) ∈ V) | |
37 | 23, 36 | ax-mp 5 | . . . . 5 ⊢ [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) ∈ V |
38 | 35, 22, 37 | fvmpt 6745 | . . . 4 ⊢ ((𝐾 · 𝑈) ∈ 𝐵 → (𝐹‘(𝐾 · 𝑈)) = [(𝐾 · 𝑈)](𝑀 ~QG 𝐺)) |
39 | 18, 38 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹‘(𝐾 · 𝑈)) = [(𝐾 · 𝑈)](𝑀 ~QG 𝐺)) |
40 | 1, 16, 4, 3 | lmodvscl 19644 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑉 ∈ 𝐵) → (𝐾 · 𝑉) ∈ 𝐵) |
41 | 5, 7, 28, 40 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐾 · 𝑉) ∈ 𝐵) |
42 | eceq1 8310 | . . . . 5 ⊢ (𝑥 = (𝐾 · 𝑉) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺)) | |
43 | ecexg 8276 | . . . . . 6 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [(𝐾 · 𝑉)](𝑀 ~QG 𝐺) ∈ V) | |
44 | 23, 43 | ax-mp 5 | . . . . 5 ⊢ [(𝐾 · 𝑉)](𝑀 ~QG 𝐺) ∈ V |
45 | 42, 22, 44 | fvmpt 6745 | . . . 4 ⊢ ((𝐾 · 𝑉) ∈ 𝐵 → (𝐹‘(𝐾 · 𝑉)) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺)) |
46 | 41, 45 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹‘(𝐾 · 𝑉)) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺)) |
47 | 39, 46 | eqeq12d 2814 | . 2 ⊢ (𝜑 → ((𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)) ↔ [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺))) |
48 | 20, 34, 47 | 3imtr4d 297 | 1 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 Er wer 8269 [cec 8270 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 /s cqus 16770 SubGrpcsubg 18265 ~QG cqg 18267 LModclmod 19627 LSubSpclss 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-ec 8274 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-eqg 18270 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-lss 19697 |
This theorem is referenced by: qusscaval 30972 quslmod 30974 quslmhm 30975 |
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