Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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.) |
Ref | Expression |
---|---|
zlmbas.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
Ref | Expression |
---|---|
zlmsca | ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaid 16748 | . . 3 ⊢ Scalar = Slot (Scalar‘ndx) | |
2 | 5re 11815 | . . . . 5 ⊢ 5 ∈ ℝ | |
3 | 5lt6 11909 | . . . . 5 ⊢ 5 < 6 | |
4 | 2, 3 | ltneii 10843 | . . . 4 ⊢ 5 ≠ 6 |
5 | scandx 16747 | . . . . 5 ⊢ (Scalar‘ndx) = 5 | |
6 | vscandx 16749 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
7 | 5, 6 | neeq12i 3001 | . . . 4 ⊢ ((Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ 5 ≠ 6) |
8 | 4, 7 | mpbir 234 | . . 3 ⊢ (Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) |
9 | 1, 8 | setsnid 16654 | . 2 ⊢ (Scalar‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (Scalar‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
10 | zringring 20304 | . . 3 ⊢ ℤring ∈ Ring | |
11 | 1 | setsid 16653 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ ℤring ∈ Ring) → ℤring = (Scalar‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉))) |
12 | 10, 11 | mpan2 691 | . 2 ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉))) |
13 | zlmbas.w | . . . 4 ⊢ 𝑊 = (ℤMod‘𝐺) | |
14 | eqid 2739 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
15 | 13, 14 | zlmval 20348 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
16 | 15 | fveq2d 6690 | . 2 ⊢ (𝐺 ∈ 𝑉 → (Scalar‘𝑊) = (Scalar‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
17 | 9, 12, 16 | 3eqtr4a 2800 | 1 ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 〈cop 4532 ‘cfv 6349 (class class class)co 7182 5c5 11786 6c6 11787 ndxcnx 16595 sSet csts 16596 Scalarcsca 16683 ·𝑠 cvsca 16684 .gcmg 18354 Ringcrg 19428 ℤringzring 20301 ℤModczlm 20333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-addf 10706 ax-mulf 10707 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-sca 16696 df-vsca 16697 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-0g 16830 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-grp 18234 df-minusg 18235 df-subg 18406 df-cmn 19038 df-mgp 19371 df-ur 19383 df-ring 19430 df-cring 19431 df-subrg 19664 df-cnfld 20230 df-zring 20302 df-zlm 20337 |
This theorem is referenced by: zlmlmod 20355 zlmassa 20728 zlmclm 23876 nmmulg 31500 cnzh 31502 rezh 31503 |
Copyright terms: Public domain | W3C validator |