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Mirrors > Home > MPE Home > Th. List > sratset | Structured version Visualization version GIF version |
Description: Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
sratset | ⊢ (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srapart.a | . 2 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
2 | srapart.s | . 2 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
3 | tsetid 17298 | . 2 ⊢ TopSet = Slot (TopSet‘ndx) | |
4 | slotstnscsi 17305 | . . . 4 ⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx)) | |
5 | 4 | simp1i 1140 | . . 3 ⊢ (TopSet‘ndx) ≠ (Scalar‘ndx) |
6 | 5 | necomi 2996 | . 2 ⊢ (Scalar‘ndx) ≠ (TopSet‘ndx) |
7 | 4 | simp2i 1141 | . . 3 ⊢ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) |
8 | 7 | necomi 2996 | . 2 ⊢ ( ·𝑠 ‘ndx) ≠ (TopSet‘ndx) |
9 | 4 | simp3i 1142 | . . 3 ⊢ (TopSet‘ndx) ≠ (·𝑖‘ndx) |
10 | 9 | necomi 2996 | . 2 ⊢ (·𝑖‘ndx) ≠ (TopSet‘ndx) |
11 | 1, 2, 3, 6, 8, 10 | sralem 20790 | 1 ⊢ (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ≠ wne 2941 ⊆ wss 3949 ‘cfv 6544 ndxcnx 17126 Basecbs 17144 Scalarcsca 17200 ·𝑠 cvsca 17201 ·𝑖cip 17202 TopSetcts 17203 subringAlg csra 20781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-sets 17097 df-slot 17115 df-ndx 17127 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-sra 20785 |
This theorem is referenced by: sratopn 20805 |
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