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Mirrors > Home > MPE Home > Th. List > frlmvscafval | Structured version Visualization version GIF version |
Description: Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
frlmvscafval.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmvscafval.b | ⊢ 𝐵 = (Base‘𝑌) |
frlmvscafval.k | ⊢ 𝐾 = (Base‘𝑅) |
frlmvscafval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
frlmvscafval.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
frlmvscafval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
frlmvscafval.v | ⊢ ∙ = ( ·𝑠 ‘𝑌) |
frlmvscafval.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
frlmvscafval | ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmvscafval.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | frlmvscafval.y | . . . . . . . 8 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
3 | frlmvscafval.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌) | |
4 | 2, 3 | frlmrcl 20903 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V) |
6 | frlmvscafval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 2, 3 | frlmpws 20896 | . . . . . 6 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
8 | 5, 6, 7 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
9 | 8 | fveq2d 6676 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
10 | frlmvscafval.v | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
11 | 3 | fvexi 6686 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | eqid 2823 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
13 | eqid 2823 | . . . . . 6 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) | |
14 | 12, 13 | ressvsca 16653 | . . . . 5 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
15 | 11, 14 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
16 | 9, 10, 15 | 3eqtr4g 2883 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))) |
17 | 16 | oveqd 7175 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋)) |
18 | eqid 2823 | . . 3 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
19 | eqid 2823 | . . 3 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
20 | frlmvscafval.t | . . . 4 ⊢ · = (.r‘𝑅) | |
21 | rlmvsca 19976 | . . . 4 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
22 | 20, 21 | eqtri 2846 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
23 | eqid 2823 | . . 3 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
24 | eqid 2823 | . . 3 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
25 | fvexd 6687 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ V) | |
26 | frlmvscafval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
27 | frlmvscafval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
28 | rlmsca 19974 | . . . . . . 7 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
29 | 5, 28 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
30 | 29 | fveq2d 6676 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
31 | 27, 30 | syl5eq 2870 | . . . 4 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
32 | 26, 31 | eleqtrd 2917 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))) |
33 | 8 | fveq2d 6676 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
34 | 3, 33 | syl5eq 2870 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
35 | 12, 19 | ressbasss 16558 | . . . . 5 ⊢ (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) |
36 | 34, 35 | eqsstrdi 4023 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
37 | 36, 1 | sseldd 3970 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
38 | 18, 19, 22, 13, 23, 24, 25, 6, 32, 37 | pwsvscafval 16769 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
39 | 17, 38 | eqtrd 2858 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 Basecbs 16485 ↾s cress 16486 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 ↑s cpws 16722 ringLModcrglmod 19943 freeLMod cfrlm 20892 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-prds 16723 df-pws 16725 df-sra 19946 df-rgmod 19947 df-dsmm 20878 df-frlm 20893 |
This theorem is referenced by: frlmvscaval 20914 uvcresum 20939 matvsca2 21039 matunitlindflem1 34890 matunitlindflem2 34891 frlmvscadiccat 39152 0prjspnrel 39276 zlmodzxzscm 44412 aacllem 44909 |
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