Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qusscaval | Structured version Visualization version GIF version |
Description: Value of the scalar multiplication operation on the quotient structure. (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) |
Ref | Expression |
---|---|
qusscaval | ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusscaval.n | . . . . 5 ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))) |
3 | eqgvscpbl.v | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑀)) |
5 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
6 | ovex 7175 | . . . . 5 ⊢ (𝑀 ~QG 𝐺) ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑀 ~QG 𝐺) ∈ V) |
8 | eqgvscpbl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
9 | 2, 4, 5, 7, 8 | qusval 16798 | . . 3 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) “s 𝑀)) |
10 | 2, 4, 5, 7, 8 | quslem 16799 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~QG 𝐺))) |
11 | eqid 2821 | . . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
12 | eqgvscpbl.s | . . 3 ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) | |
13 | eqgvscpbl.p | . . 3 ⊢ · = ( ·𝑠 ‘𝑀) | |
14 | qusscaval.m | . . 3 ⊢ ∙ = ( ·𝑠 ‘𝑁) | |
15 | eqgvscpbl.e | . . . 4 ⊢ ∼ = (𝑀 ~QG 𝐺) | |
16 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑀 ∈ LMod) |
17 | eqgvscpbl.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) | |
18 | 17 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ (LSubSp‘𝑀)) |
19 | simpr1 1190 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆) | |
20 | simpr2 1191 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵) | |
21 | simpr3 1192 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵) | |
22 | 3, 15, 12, 13, 16, 18, 19, 1, 14, 5, 20, 21 | qusvscpbl 30927 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑢) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑣) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝑘 · 𝑢)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝑘 · 𝑣)))) |
23 | 9, 4, 10, 8, 11, 12, 13, 14, 22 | imasvscaval 16794 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋))) |
24 | eceq1 8313 | . . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~QG 𝐺) = [𝑋](𝑀 ~QG 𝐺)) | |
25 | ecexg 8279 | . . . . . 6 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [𝑋](𝑀 ~QG 𝐺) ∈ V) | |
26 | 6, 25 | ax-mp 5 | . . . . 5 ⊢ [𝑋](𝑀 ~QG 𝐺) ∈ V |
27 | 24, 5, 26 | fvmpt 6754 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) |
28 | 27 | 3ad2ant3 1131 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) |
29 | 28 | oveq2d 7158 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺))) |
30 | 3, 11, 13, 12 | lmodvscl 19634 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) |
31 | 8, 30 | syl3an1 1159 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) |
32 | eceq1 8313 | . . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | |
33 | ecexg 8279 | . . . . 5 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [(𝐾 · 𝑋)](𝑀 ~QG 𝐺) ∈ V) | |
34 | 6, 33 | ax-mp 5 | . . . 4 ⊢ [(𝐾 · 𝑋)](𝑀 ~QG 𝐺) ∈ V |
35 | 32, 5, 34 | fvmpt 6754 | . . 3 ⊢ ((𝐾 · 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
36 | 31, 35 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
37 | 23, 29, 36 | 3eqtr3d 2864 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ↦ cmpt 5132 ‘cfv 6341 (class class class)co 7142 [cec 8273 / cqs 8274 Basecbs 16466 Scalarcsca 16551 ·𝑠 cvsca 16552 /s cqus 16761 ~QG cqg 18258 LModclmod 19617 LSubSpclss 19686 |
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-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-ec 8277 df-qs 8281 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-0g 16698 df-imas 16764 df-qus 16765 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-grp 18089 df-minusg 18090 df-sbg 18091 df-subg 18259 df-eqg 18261 df-mgp 19223 df-ur 19235 df-ring 19282 df-lmod 19619 df-lss 19687 |
This theorem is referenced by: (None) |
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