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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 7464 | . . . 4 ⊢ (𝐺 sSet 〈(Scalar‘ndx), ℤring〉) ∈ V | |
| 2 | zlmvsca.2 | . . . . 5 ⊢ · = (.g‘𝐺) | |
| 3 | 2 | fvexi 6920 | . . . 4 ⊢ · ∈ V | 
| 4 | vscaid 17364 | . . . . 5 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
| 5 | 4 | setsid 17244 | . . . 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 21526 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) | 
| 9 | 8 | fveq2d 6910 | . . 3 ⊢ (𝐺 ∈ V → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉))) | 
| 10 | 6, 9 | eqtr4id 2796 | . 2 ⊢ (𝐺 ∈ V → · = ( ·𝑠 ‘𝑊)) | 
| 11 | 4 | str0 17226 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) | 
| 12 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅) | |
| 13 | 2, 12 | eqtrid 2789 | . . 3 ⊢ (¬ 𝐺 ∈ V → · = ∅) | 
| 14 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (ℤMod‘𝐺) = ∅) | |
| 15 | 7, 14 | eqtrid 2789 | . . . 4 ⊢ (¬ 𝐺 ∈ V → 𝑊 = ∅) | 
| 16 | 15 | fveq2d 6910 | . . 3 ⊢ (¬ 𝐺 ∈ V → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘∅)) | 
| 17 | 11, 13, 16 | 3eqtr4a 2803 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 Scalarcsca 17300 ·𝑠 cvsca 17301 .gcmg 19085 ℤringczring 21457 ℤModczlm 21511 | 
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 | 
| 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-ral 3062 df-rex 3071 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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-sets 17201 df-slot 17219 df-ndx 17231 df-vsca 17314 df-zlm 21515 | 
| This theorem is referenced by: zlmlmod 21537 zlmassa 21923 clmzlmvsca 25146 nmmulg 33967 cnzh 33969 rezh 33970 | 
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