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Mirrors > Home > MPE Home > Th. List > zlmvsca | Structured version Visualization version GIF version |
Description: Scalar multiplication operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
zlmbas.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmvsca.2 | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
zlmvsca | ⊢ · = ( ·𝑠 ‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7251 | . . . 4 ⊢ (𝐺 sSet 〈(Scalar‘ndx), ℤring〉) ∈ V | |
2 | zlmvsca.2 | . . . . 5 ⊢ · = (.g‘𝐺) | |
3 | 2 | fvexi 6736 | . . . 4 ⊢ · ∈ V |
4 | vscaid 16864 | . . . . 5 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
5 | 4 | setsid 16763 | . . . 4 ⊢ (((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) ∈ V ∧ · ∈ V) → · = ( ·𝑠 ‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉))) |
6 | 1, 3, 5 | mp2an 692 | . . 3 ⊢ · = ( ·𝑠 ‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) |
7 | zlmbas.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
8 | 7, 2 | zlmval 20487 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) |
9 | 8 | fveq2d 6726 | . . 3 ⊢ (𝐺 ∈ V → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉))) |
10 | 6, 9 | eqtr4id 2797 | . 2 ⊢ (𝐺 ∈ V → · = ( ·𝑠 ‘𝑊)) |
11 | 4 | str0 16747 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) |
12 | fvprc 6714 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅) | |
13 | 2, 12 | syl5eq 2790 | . . 3 ⊢ (¬ 𝐺 ∈ V → · = ∅) |
14 | fvprc 6714 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (ℤMod‘𝐺) = ∅) | |
15 | 7, 14 | syl5eq 2790 | . . . 4 ⊢ (¬ 𝐺 ∈ V → 𝑊 = ∅) |
16 | 15 | fveq2d 6726 | . . 3 ⊢ (¬ 𝐺 ∈ V → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘∅)) |
17 | 11, 13, 16 | 3eqtr4a 2804 | . 2 ⊢ (¬ 𝐺 ∈ V → · = ( ·𝑠 ‘𝑊)) |
18 | 10, 17 | pm2.61i 185 | 1 ⊢ · = ( ·𝑠 ‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 Vcvv 3413 ∅c0 4242 〈cop 4552 ‘cfv 6385 (class class class)co 7218 sSet csts 16721 ndxcnx 16749 Scalarcsca 16810 ·𝑠 cvsca 16811 .gcmg 18493 ℤringzring 20440 ℤModczlm 20472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-1cn 10792 ax-addcl 10794 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-ov 7221 df-oprab 7222 df-mpo 7223 df-om 7650 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-sets 16722 df-slot 16740 df-ndx 16750 df-vsca 16824 df-zlm 20476 |
This theorem is referenced by: zlmlmod 20494 zlmassa 20867 clmzlmvsca 24015 nmmulg 31635 cnzh 31637 rezh 31638 |
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