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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 7394 | . . . 4 ⊢ (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) ∈ V | |
2 | zlmvsca.2 | . . . . 5 ⊢ · = (.g‘𝐺) | |
3 | 2 | fvexi 6860 | . . . 4 ⊢ · ∈ V |
4 | vscaid 17209 | . . . . 5 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
5 | 4 | setsid 17088 | . . . 4 ⊢ (((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) ∈ V ∧ · ∈ V) → · = ( ·𝑠 ‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))) |
6 | 1, 3, 5 | mp2an 691 | . . 3 ⊢ · = ( ·𝑠 ‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩)) |
7 | zlmbas.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
8 | 7, 2 | zlmval 20939 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩)) |
9 | 8 | fveq2d 6850 | . . 3 ⊢ (𝐺 ∈ V → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))) |
10 | 6, 9 | eqtr4id 2792 | . 2 ⊢ (𝐺 ∈ V → · = ( ·𝑠 ‘𝑊)) |
11 | 4 | str0 17069 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) |
12 | fvprc 6838 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅) | |
13 | 2, 12 | eqtrid 2785 | . . 3 ⊢ (¬ 𝐺 ∈ V → · = ∅) |
14 | fvprc 6838 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (ℤMod‘𝐺) = ∅) | |
15 | 7, 14 | eqtrid 2785 | . . . 4 ⊢ (¬ 𝐺 ∈ V → 𝑊 = ∅) |
16 | 15 | fveq2d 6850 | . . 3 ⊢ (¬ 𝐺 ∈ V → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘∅)) |
17 | 11, 13, 16 | 3eqtr4a 2799 | . 2 ⊢ (¬ 𝐺 ∈ V → · = ( ·𝑠 ‘𝑊)) |
18 | 10, 17 | pm2.61i 182 | 1 ⊢ · = ( ·𝑠 ‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 ⟨cop 4596 ‘cfv 6500 (class class class)co 7361 sSet csts 17043 ndxcnx 17073 Scalarcsca 17144 ·𝑠 cvsca 17145 .gcmg 18880 ℤringczring 20892 ℤModczlm 20924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-1cn 11117 ax-addcl 11119 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-sets 17044 df-slot 17062 df-ndx 17074 df-vsca 17158 df-zlm 20928 |
This theorem is referenced by: zlmlmod 20950 zlmassa 21328 clmzlmvsca 24499 nmmulg 32613 cnzh 32615 rezh 32616 |
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