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 2821 | . . . 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 30919 | . . 3 ⊢ (𝜑 → (𝑈(𝑀 ~QG 𝐺)𝑉 → (𝐾 · 𝑈)(𝑀 ~QG 𝐺)(𝐾 · 𝑉))) |
9 | eqid 2821 | . . . . . . 7 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
10 | 9 | lsssubg 19729 | . . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀)) |
11 | 5, 6, 10 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀)) |
12 | 1, 2 | eqger 18330 | . . . . 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 8338 | . . 3 ⊢ (𝜑 → (𝑈(𝑀 ~QG 𝐺)𝑉 ↔ [𝑈](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺))) |
16 | eqid 2821 | . . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
17 | 1, 16, 4, 3 | lmodvscl 19651 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑈 ∈ 𝐵) → (𝐾 · 𝑈) ∈ 𝐵) |
18 | 5, 7, 14, 17 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐾 · 𝑈) ∈ 𝐵) |
19 | 13, 18 | erth 8338 | . . 3 ⊢ (𝜑 → ((𝐾 · 𝑈)(𝑀 ~QG 𝐺)(𝐾 · 𝑉) ↔ [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺))) |
20 | 8, 15, 19 | 3imtr3d 295 | . 2 ⊢ (𝜑 → ([𝑈](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺) → [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺))) |
21 | eceq1 8327 | . . . . 5 ⊢ (𝑥 = 𝑈 → [𝑥](𝑀 ~QG 𝐺) = [𝑈](𝑀 ~QG 𝐺)) | |
22 | qusvscpbl.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
23 | ovex 7189 | . . . . . 6 ⊢ (𝑀 ~QG 𝐺) ∈ V | |
24 | ecexg 8293 | . . . . . 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 6768 | . . . 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 8327 | . . . . 5 ⊢ (𝑥 = 𝑉 → [𝑥](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺)) | |
30 | ecexg 8293 | . . . . . 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 6768 | . . . 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 2837 | . 2 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ [𝑈](𝑀 ~QG 𝐺) = [𝑉](𝑀 ~QG 𝐺))) |
35 | eceq1 8327 | . . . . 5 ⊢ (𝑥 = (𝐾 · 𝑈) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑈)](𝑀 ~QG 𝐺)) | |
36 | ecexg 8293 | . . . . . 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 6768 | . . . 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 19651 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑉 ∈ 𝐵) → (𝐾 · 𝑉) ∈ 𝐵) |
41 | 5, 7, 28, 40 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐾 · 𝑉) ∈ 𝐵) |
42 | eceq1 8327 | . . . . 5 ⊢ (𝑥 = (𝐾 · 𝑉) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺)) | |
43 | ecexg 8293 | . . . . . 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 6768 | . . . 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 2837 | . 2 ⊢ (𝜑 → ((𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)) ↔ [(𝐾 · 𝑈)](𝑀 ~QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~QG 𝐺))) |
48 | 20, 34, 47 | 3imtr4d 296 | 1 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 Er wer 8286 [cec 8287 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 /s cqus 16778 SubGrpcsubg 18273 ~QG cqg 18275 LModclmod 19634 LSubSpclss 19703 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-ec 8291 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-eqg 18278 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 |
This theorem is referenced by: qusscaval 30921 quslmod 30923 quslmhm 30924 |
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