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Mirrors > Home > MPE Home > Th. List > tnglem | Structured version Visualization version GIF version |
Description: Lemma for tngbas 24150 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.) |
Ref | Expression |
---|---|
tngbas.t | β’ π = (πΊ toNrmGrp π) |
tnglem.e | β’ πΈ = Slot (πΈβndx) |
tnglem.t | β’ (πΈβndx) β (TopSetβndx) |
tnglem.d | β’ (πΈβndx) β (distβndx) |
Ref | Expression |
---|---|
tnglem | β’ (π β π β (πΈβπΊ) = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tnglem.e | . . . . 5 β’ πΈ = Slot (πΈβndx) | |
2 | tnglem.d | . . . . 5 β’ (πΈβndx) β (distβndx) | |
3 | 1, 2 | setsnid 17141 | . . . 4 β’ (πΈβπΊ) = (πΈβ(πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©)) |
4 | tnglem.t | . . . . 5 β’ (πΈβndx) β (TopSetβndx) | |
5 | 1, 4 | setsnid 17141 | . . . 4 β’ (πΈβ(πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©)) = (πΈβ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©)) |
6 | 3, 5 | eqtri 2760 | . . 3 β’ (πΈβπΊ) = (πΈβ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©)) |
7 | tngbas.t | . . . . 5 β’ π = (πΊ toNrmGrp π) | |
8 | eqid 2732 | . . . . 5 β’ (-gβπΊ) = (-gβπΊ) | |
9 | eqid 2732 | . . . . 5 β’ (π β (-gβπΊ)) = (π β (-gβπΊ)) | |
10 | eqid 2732 | . . . . 5 β’ (MetOpenβ(π β (-gβπΊ))) = (MetOpenβ(π β (-gβπΊ))) | |
11 | 7, 8, 9, 10 | tngval 24147 | . . . 4 β’ ((πΊ β V β§ π β π) β π = ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©)) |
12 | 11 | fveq2d 6895 | . . 3 β’ ((πΊ β V β§ π β π) β (πΈβπ) = (πΈβ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©))) |
13 | 6, 12 | eqtr4id 2791 | . 2 β’ ((πΊ β V β§ π β π) β (πΈβπΊ) = (πΈβπ)) |
14 | 1 | str0 17121 | . . . . 5 β’ β = (πΈββ ) |
15 | 14 | eqcomi 2741 | . . . 4 β’ (πΈββ ) = β |
16 | reldmtng 24146 | . . . 4 β’ Rel dom toNrmGrp | |
17 | 15, 7, 16 | oveqprc 17124 | . . 3 β’ (Β¬ πΊ β V β (πΈβπΊ) = (πΈβπ)) |
18 | 17 | adantr 481 | . 2 β’ ((Β¬ πΊ β V β§ π β π) β (πΈβπΊ) = (πΈβπ)) |
19 | 13, 18 | pm2.61ian 810 | 1 β’ (π β π β (πΈβπΊ) = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 Vcvv 3474 β c0 4322 β¨cop 4634 β ccom 5680 βcfv 6543 (class class class)co 7408 sSet csts 17095 Slot cslot 17113 ndxcnx 17125 TopSetcts 17202 distcds 17205 -gcsg 18820 MetOpencmopn 20933 toNrmGrp ctng 24086 |
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-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-res 5688 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-sets 17096 df-slot 17114 df-tng 24092 |
This theorem is referenced by: tngbas 24150 tngplusg 24152 tngmulr 24155 tngsca 24157 tngvsca 24159 tngip 24161 |
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