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| Mirrors > Home > MPE Home > Th. List > Mathboxes > qusvsval | 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) | 
| qusvsval.n | ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) | 
| qusvsval.m | ⊢ ∙ = ( ·𝑠 ‘𝑁) | 
| qusvsval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| qusvsval | ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqgvscpbl.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑆) | |
| 2 | qusvsval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | qusvsval.n | . . . . . 6 ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) | |
| 4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))) | 
| 5 | eqgvscpbl.v | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑀)) | 
| 7 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
| 8 | ovex 7464 | . . . . . 6 ⊢ (𝑀 ~QG 𝐺) ∈ V | |
| 9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑀 ~QG 𝐺) ∈ V) | 
| 10 | eqgvscpbl.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
| 11 | 4, 6, 7, 9, 10 | qusval 17587 | . . . 4 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) “s 𝑀)) | 
| 12 | 4, 6, 7, 9, 10 | quslem 17588 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~QG 𝐺))) | 
| 13 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
| 14 | eqgvscpbl.s | . . . 4 ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) | |
| 15 | eqgvscpbl.p | . . . 4 ⊢ · = ( ·𝑠 ‘𝑀) | |
| 16 | qusvsval.m | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑁) | |
| 17 | eqgvscpbl.e | . . . . 5 ⊢ ∼ = (𝑀 ~QG 𝐺) | |
| 18 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑀 ∈ LMod) | 
| 19 | eqgvscpbl.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) | |
| 20 | 19 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ (LSubSp‘𝑀)) | 
| 21 | simpr1 1195 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆) | |
| 22 | simpr2 1196 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵) | |
| 23 | simpr3 1197 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵) | |
| 24 | 5, 17, 14, 15, 18, 20, 21, 3, 16, 7, 22, 23 | qusvscpbl 33379 | . . . 4 ⊢ ((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)))) | 
| 25 | 11, 6, 12, 10, 13, 14, 15, 16, 24 | imasvscaval 17583 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋))) | 
| 26 | 1, 2, 25 | mpd3an23 1465 | . 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋))) | 
| 27 | eceq1 8784 | . . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~QG 𝐺) = [𝑋](𝑀 ~QG 𝐺)) | |
| 28 | ecexg 8749 | . . . . . 6 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [𝑋](𝑀 ~QG 𝐺) ∈ V) | |
| 29 | 8, 28 | ax-mp 5 | . . . . 5 ⊢ [𝑋](𝑀 ~QG 𝐺) ∈ V | 
| 30 | 27, 7, 29 | fvmpt 7016 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) | 
| 31 | 2, 30 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) | 
| 32 | 31 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺))) | 
| 33 | 5, 13, 15, 14 | lmodvscl 20876 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) | 
| 34 | 10, 1, 2, 33 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝐾 · 𝑋) ∈ 𝐵) | 
| 35 | eceq1 8784 | . . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | |
| 36 | ecexg 8749 | . . . . 5 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [(𝐾 · 𝑋)](𝑀 ~QG 𝐺) ∈ V) | |
| 37 | 8, 36 | ax-mp 5 | . . . 4 ⊢ [(𝐾 · 𝑋)](𝑀 ~QG 𝐺) ∈ V | 
| 38 | 35, 7, 37 | fvmpt 7016 | . . 3 ⊢ ((𝐾 · 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | 
| 39 | 34, 38 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | 
| 40 | 26, 32, 39 | 3eqtr3d 2785 | 1 ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 [cec 8743 / cqs 8744 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 /s cqus 17550 ~QG cqg 19140 LModclmod 20858 LSubSpclss 20929 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-lmod 20860 df-lss 20930 | 
| This theorem is referenced by: lmhmqusker 33445 | 
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