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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 2994 | . . 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 21310 | . . 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 2731 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
10 | 8, 9 | zlmval 21376 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
11 | 10 | fveq2d 6895 | . 2 ⊢ (𝐺 ∈ 𝑉 → (Scalar‘𝑊) = (Scalar‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
12 | 4, 7, 11 | 3eqtr4a 2797 | 1 ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 〈cop 4634 ‘cfv 6543 (class class class)co 7412 sSet csts 17103 ndxcnx 17133 Scalarcsca 17207 ·𝑠 cvsca 17208 .gcmg 18993 Ringcrg 20134 ℤringczring 21307 ℤModczlm 21361 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 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 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-subrng 20442 df-subrg 20467 df-cnfld 21235 df-zring 21308 df-zlm 21365 |
This theorem is referenced by: zlmlmod 21387 zlmassa 21767 zlmclm 24960 nmmulg 33414 cnzh 33416 rezh 33417 |
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