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 2820 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
6 | ovex 7182 | . . . . 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 16810 | . . 3 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) “s 𝑀)) |
10 | 2, 4, 5, 7, 8 | quslem 16811 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~QG 𝐺))) |
11 | eqid 2820 | . . 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 1189 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆) | |
20 | simpr2 1190 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵) | |
21 | simpr3 1191 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵) | |
22 | 3, 15, 12, 13, 16, 18, 19, 1, 14, 5, 20, 21 | qusvscpbl 30941 | . . 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 16806 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋))) |
24 | eceq1 8320 | . . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~QG 𝐺) = [𝑋](𝑀 ~QG 𝐺)) | |
25 | ecexg 8286 | . . . . . 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 6761 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) |
28 | 27 | 3ad2ant3 1130 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) |
29 | 28 | oveq2d 7165 | . 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 19646 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) |
31 | 8, 30 | syl3an1 1158 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) |
32 | eceq1 8320 | . . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | |
33 | ecexg 8286 | . . . . 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 6761 | . . 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 2863 | 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 1082 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ↦ cmpt 5139 ‘cfv 6348 (class class class)co 7149 [cec 8280 / cqs 8281 Basecbs 16478 Scalarcsca 16563 ·𝑠 cvsca 16564 /s cqus 16773 ~QG cqg 18270 LModclmod 19629 LSubSpclss 19698 |
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 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-ec 8284 df-qs 8288 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-0g 16710 df-imas 16776 df-qus 16777 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-eqg 18273 df-mgp 19235 df-ur 19247 df-ring 19294 df-lmod 19631 df-lss 19699 |
This theorem is referenced by: (None) |
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