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Mirrors > Home > MPE Home > Th. List > zlmsca | Structured version Visualization version GIF version |
Description: Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.) |
Ref | Expression |
---|---|
zlmbas.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
Ref | Expression |
---|---|
zlmsca | ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaid 17267 | . . 3 ⊢ Scalar = Slot (Scalar‘ndx) | |
2 | vscandxnscandx 17276 | . . . 4 ⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx) | |
3 | 2 | necomi 2989 | . . 3 ⊢ (Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) |
4 | 1, 3 | setsnid 17149 | . 2 ⊢ (Scalar‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) = (Scalar‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
5 | zringring 21332 | . . 3 ⊢ ℤring ∈ Ring | |
6 | 1 | setsid 17148 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ ℤring ∈ Ring) → ℤring = (Scalar‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))) |
7 | 5, 6 | mpan2 688 | . 2 ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))) |
8 | zlmbas.w | . . . 4 ⊢ 𝑊 = (ℤMod‘𝐺) | |
9 | eqid 2726 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
10 | 8, 9 | zlmval 21398 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
11 | 10 | fveq2d 6888 | . 2 ⊢ (𝐺 ∈ 𝑉 → (Scalar‘𝑊) = (Scalar‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩))) |
12 | 4, 7, 11 | 3eqtr4a 2792 | 1 ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⟨cop 4629 ‘cfv 6536 (class class class)co 7404 sSet csts 17103 ndxcnx 17133 Scalarcsca 17207 ·𝑠 cvsca 17208 .gcmg 18993 Ringcrg 20136 ℤringczring 21329 ℤModczlm 21383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-subg 19048 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-subrng 20444 df-subrg 20469 df-cnfld 21237 df-zring 21330 df-zlm 21387 |
This theorem is referenced by: zlmlmod 21409 zlmassa 21793 zlmclm 24990 nmmulg 33478 cnzh 33480 rezh 33481 |
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