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Mirrors > Home > MPE Home > Th. List > tnglem | Structured version Visualization version GIF version |
Description: Lemma for tngbas 24021 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 17089 | . . . 4 β’ (πΈβπΊ) = (πΈβ(πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©)) |
4 | tnglem.t | . . . . 5 β’ (πΈβndx) β (TopSetβndx) | |
5 | 1, 4 | setsnid 17089 | . . . 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 2761 | . . 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 2733 | . . . . 5 β’ (-gβπΊ) = (-gβπΊ) | |
9 | eqid 2733 | . . . . 5 β’ (π β (-gβπΊ)) = (π β (-gβπΊ)) | |
10 | eqid 2733 | . . . . 5 β’ (MetOpenβ(π β (-gβπΊ))) = (MetOpenβ(π β (-gβπΊ))) | |
11 | 7, 8, 9, 10 | tngval 24018 | . . . 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 6850 | . . 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 2792 | . 2 β’ ((πΊ β V β§ π β π) β (πΈβπΊ) = (πΈβπ)) |
14 | 1 | str0 17069 | . . . . 5 β’ β = (πΈββ ) |
15 | 14 | eqcomi 2742 | . . . 4 β’ (πΈββ ) = β |
16 | reldmtng 24017 | . . . 4 β’ Rel dom toNrmGrp | |
17 | 15, 7, 16 | oveqprc 17072 | . . 3 β’ (Β¬ πΊ β V β (πΈβπΊ) = (πΈβπ)) |
18 | 17 | adantr 482 | . 2 β’ ((Β¬ πΊ β V β§ π β π) β (πΈβπΊ) = (πΈβπ)) |
19 | 13, 18 | pm2.61ian 811 | 1 β’ (π β π β (πΈβπΊ) = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 Vcvv 3447 β c0 4286 β¨cop 4596 β ccom 5641 βcfv 6500 (class class class)co 7361 sSet csts 17043 Slot cslot 17061 ndxcnx 17073 TopSetcts 17147 distcds 17150 -gcsg 18758 MetOpencmopn 20809 toNrmGrp ctng 23957 |
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-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-res 5649 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-sets 17044 df-slot 17062 df-tng 23963 |
This theorem is referenced by: tngbas 24021 tngplusg 24023 tngmulr 24026 tngsca 24028 tngvsca 24030 tngip 24032 |
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