Theorem sratsetg 14610

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.)

Hypotheses

Ref	Expression
srapart.a	|- ( ph -> A = ( (subringAlg ` W ) ` S ) )
srapart.s	|- ( ph -> S C_ ( Base ` W ) )
srapart.ex	|- ( ph -> W e. X )

Assertion

Ref	Expression
sratsetg	|- ( ph -> (TopSet ` W ) = (TopSet ` A ) )

Proof of Theorem sratsetg

Step	Hyp	Ref	Expression
1		srapart.a	. 2 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
2		srapart.s	. 2 |- ( ph -> S C_ ( Base ` W ) )
3		srapart.ex	. 2 |- ( ph -> W e. X )
4		tsetslid 13418	. 2 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
5		slotstnscsi 13425	. . . 4 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )
6	5	simp1i 1033	. . 3 |- (TopSet ` ndx ) =/= (Scalar ` ndx )
7	6	necomi 2499	. 2 |- (Scalar ` ndx ) =/= (TopSet ` ndx )
8	5	simp2i 1034	. . 3 |- (TopSet ` ndx ) =/= ( .s ` ndx )
9	8	necomi 2499	. 2 |- ( .s ` ndx ) =/= (TopSet ` ndx )
10	5	simp3i 1035	. . 3 |- (TopSet ` ndx ) =/= ( .i ` ndx )
11	10	necomi 2499	. 2 |- ( .i ` ndx ) =/= (TopSet ` ndx )
12	1, 2, 3, 4, 7, 9, 11	sralemg 14603	1 |- ( ph -> (TopSet ` W ) = (TopSet ` A ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   =/= wne 2414   C_ wss 3213   ` cfv 5354   ndxcnx 13226   Basecbs 13229  Scalarcsca 13310   .scvsca 13311   .icip 13312  TopSetcts 13313  subringAlg csra 14598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-mulr 13321  df-sca 13323  df-vsca 13324  df-ip 13325  df-tset 13326  df-sra 14600

This theorem is referenced by:  sratopng  14612