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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmtset | Structured version Visualization version GIF version |
Description: Topology in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 12-Nov-2024.) |
Ref | Expression |
---|---|
zlmlem2.1 | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmtset.1 | ⊢ 𝐽 = (TopSet‘𝐺) |
Ref | Expression |
---|---|
zlmtset | ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmtset.1 | . . 3 ⊢ 𝐽 = (TopSet‘𝐺) | |
2 | tsetid 17302 | . . . 4 ⊢ TopSet = Slot (TopSet‘ndx) | |
3 | slotstnscsi 17309 | . . . . 5 ⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx)) | |
4 | 3 | simp1i 1139 | . . . 4 ⊢ (TopSet‘ndx) ≠ (Scalar‘ndx) |
5 | 2, 4 | setsnid 17146 | . . 3 ⊢ (TopSet‘𝐺) = (TopSet‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) |
6 | 3 | simp2i 1140 | . . . 4 ⊢ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) |
7 | 2, 6 | setsnid 17146 | . . 3 ⊢ (TopSet‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) = (TopSet‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
8 | 1, 5, 7 | 3eqtri 2764 | . 2 ⊢ 𝐽 = (TopSet‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
9 | zlmlem2.1 | . . . 4 ⊢ 𝑊 = (ℤMod‘𝐺) | |
10 | eqid 2732 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
11 | 9, 10 | zlmval 21284 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
12 | 11 | fveq2d 6895 | . 2 ⊢ (𝐺 ∈ 𝑉 → (TopSet‘𝑊) = (TopSet‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩))) |
13 | 8, 12 | eqtr4id 2791 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⟨cop 4634 ‘cfv 6543 (class class class)co 7411 sSet csts 17100 ndxcnx 17130 Scalarcsca 17204 ·𝑠 cvsca 17205 ·𝑖cip 17206 TopSetcts 17207 .gcmg 18986 ℤringczring 21217 ℤModczlm 21269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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-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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-sets 17101 df-slot 17119 df-ndx 17131 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-zlm 21273 |
This theorem is referenced by: zhmnrg 33233 |
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