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Mirrors > Home > MPE Home > Th. List > tnglemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of tnglem 23902 as of 31-Oct-2024. Lemma for tngbas 23904 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
tngbas.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tnglemOLD.2 | ⊢ 𝐸 = Slot 𝐾 |
tnglemOLD.3 | ⊢ 𝐾 ∈ ℕ |
tnglemOLD.4 | ⊢ 𝐾 < 9 |
Ref | Expression |
---|---|
tnglemOLD | ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tnglemOLD.2 | . . . . . 6 ⊢ 𝐸 = Slot 𝐾 | |
2 | tnglemOLD.3 | . . . . . 6 ⊢ 𝐾 ∈ ℕ | |
3 | 1, 2 | ndxid 16995 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 16994 | . . . . . . . 8 ⊢ (𝐸‘ndx) = 𝐾 |
5 | 2 | nnrei 12083 | . . . . . . . 8 ⊢ 𝐾 ∈ ℝ |
6 | 4, 5 | eqeltri 2833 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ ℝ |
7 | tnglemOLD.4 | . . . . . . . . 9 ⊢ 𝐾 < 9 | |
8 | 4, 7 | eqbrtri 5113 | . . . . . . . 8 ⊢ (𝐸‘ndx) < 9 |
9 | 1nn 12085 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
10 | 2nn0 12351 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
11 | 9nn0 12358 | . . . . . . . . 9 ⊢ 9 ∈ ℕ0 | |
12 | 9lt10 12669 | . . . . . . . . 9 ⊢ 9 < ;10 | |
13 | 9, 10, 11, 12 | declti 12576 | . . . . . . . 8 ⊢ 9 < ;12 |
14 | 9re 12173 | . . . . . . . . 9 ⊢ 9 ∈ ℝ | |
15 | 1nn0 12350 | . . . . . . . . . . 11 ⊢ 1 ∈ ℕ0 | |
16 | 15, 10 | deccl 12553 | . . . . . . . . . 10 ⊢ ;12 ∈ ℕ0 |
17 | 16 | nn0rei 12345 | . . . . . . . . 9 ⊢ ;12 ∈ ℝ |
18 | 6, 14, 17 | lttri 11202 | . . . . . . . 8 ⊢ (((𝐸‘ndx) < 9 ∧ 9 < ;12) → (𝐸‘ndx) < ;12) |
19 | 8, 13, 18 | mp2an 689 | . . . . . . 7 ⊢ (𝐸‘ndx) < ;12 |
20 | 6, 19 | ltneii 11189 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ ;12 |
21 | dsndx 17192 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
22 | 20, 21 | neeqtrri 3014 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (dist‘ndx) |
23 | 3, 22 | setsnid 17007 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) |
24 | 6, 8 | ltneii 11189 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 9 |
25 | tsetndx 17159 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
26 | 24, 25 | neeqtrri 3014 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) |
27 | 3, 26 | setsnid 17007 | . . . 4 ⊢ (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
28 | 23, 27 | eqtri 2764 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
29 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
30 | eqid 2736 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
31 | eqid 2736 | . . . . 5 ⊢ (𝑁 ∘ (-g‘𝐺)) = (𝑁 ∘ (-g‘𝐺)) | |
32 | eqid 2736 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ (-g‘𝐺))) = (MetOpen‘(𝑁 ∘ (-g‘𝐺))) | |
33 | 29, 30, 31, 32 | tngval 23901 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
34 | 33 | fveq2d 6829 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑇) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉))) |
35 | 28, 34 | eqtr4id 2795 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
36 | 1 | str0 16987 | . . 3 ⊢ ∅ = (𝐸‘∅) |
37 | fvprc 6817 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = ∅) | |
38 | 37 | adantr 481 | . . 3 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = ∅) |
39 | reldmtng 23900 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
40 | 39 | ovprc1 7376 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
41 | 40 | adantr 481 | . . . . 5 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐺 toNrmGrp 𝑁) = ∅) |
42 | 29, 41 | eqtrid 2788 | . . . 4 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ∅) |
43 | 42 | fveq2d 6829 | . . 3 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑇) = (𝐸‘∅)) |
44 | 36, 38, 43 | 3eqtr4a 2802 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
45 | 35, 44 | pm2.61ian 809 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ∅c0 4269 〈cop 4579 class class class wbr 5092 ∘ ccom 5624 ‘cfv 6479 (class class class)co 7337 ℝcr 10971 1c1 10973 < clt 11110 ℕcn 12074 2c2 12129 9c9 12136 ;cdc 12538 sSet csts 16961 Slot cslot 16979 ndxcnx 16991 TopSetcts 17065 distcds 17068 -gcsg 18675 MetOpencmopn 20693 toNrmGrp ctng 23840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-sets 16962 df-slot 16980 df-ndx 16992 df-tset 17078 df-ds 17081 df-tng 23846 |
This theorem is referenced by: tngbasOLD 23905 tngplusgOLD 23907 tngmulrOLD 23910 tngscaOLD 23912 tngvscaOLD 23914 tngipOLD 23916 |
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