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Mirrors > Home > MPE Home > Th. List > tnglem | Structured version Visualization version GIF version |
Description: Lemma for tngbas 22617 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
tngbas.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tnglem.2 | ⊢ 𝐸 = Slot 𝐾 |
tnglem.3 | ⊢ 𝐾 ∈ ℕ |
tnglem.4 | ⊢ 𝐾 < 9 |
Ref | Expression |
---|---|
tnglem | ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
2 | eqid 2748 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2748 | . . . . 5 ⊢ (𝑁 ∘ (-g‘𝐺)) = (𝑁 ∘ (-g‘𝐺)) | |
4 | eqid 2748 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ (-g‘𝐺))) = (MetOpen‘(𝑁 ∘ (-g‘𝐺))) | |
5 | 1, 2, 3, 4 | tngval 22615 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
6 | 5 | fveq2d 6344 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑇) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉))) |
7 | tnglem.2 | . . . . . 6 ⊢ 𝐸 = Slot 𝐾 | |
8 | tnglem.3 | . . . . . 6 ⊢ 𝐾 ∈ ℕ | |
9 | 7, 8 | ndxid 16056 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) |
10 | 7, 8 | ndxarg 16055 | . . . . . . . 8 ⊢ (𝐸‘ndx) = 𝐾 |
11 | 8 | nnrei 11192 | . . . . . . . 8 ⊢ 𝐾 ∈ ℝ |
12 | 10, 11 | eqeltri 2823 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ ℝ |
13 | tnglem.4 | . . . . . . . . 9 ⊢ 𝐾 < 9 | |
14 | 10, 13 | eqbrtri 4813 | . . . . . . . 8 ⊢ (𝐸‘ndx) < 9 |
15 | 1nn 11194 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
16 | 2nn0 11472 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
17 | 9nn0 11479 | . . . . . . . . 9 ⊢ 9 ∈ ℕ0 | |
18 | 9lt10 11836 | . . . . . . . . 9 ⊢ 9 < ;10 | |
19 | 15, 16, 17, 18 | declti 11709 | . . . . . . . 8 ⊢ 9 < ;12 |
20 | 9re 11270 | . . . . . . . . 9 ⊢ 9 ∈ ℝ | |
21 | 1nn0 11471 | . . . . . . . . . . 11 ⊢ 1 ∈ ℕ0 | |
22 | 21, 16 | deccl 11675 | . . . . . . . . . 10 ⊢ ;12 ∈ ℕ0 |
23 | 22 | nn0rei 11466 | . . . . . . . . 9 ⊢ ;12 ∈ ℝ |
24 | 12, 20, 23 | lttri 10326 | . . . . . . . 8 ⊢ (((𝐸‘ndx) < 9 ∧ 9 < ;12) → (𝐸‘ndx) < ;12) |
25 | 14, 19, 24 | mp2an 710 | . . . . . . 7 ⊢ (𝐸‘ndx) < ;12 |
26 | 12, 25 | ltneii 10313 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ ;12 |
27 | dsndx 16235 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
28 | 26, 27 | neeqtrri 2993 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (dist‘ndx) |
29 | 9, 28 | setsnid 16088 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) |
30 | 12, 14 | ltneii 10313 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 9 |
31 | tsetndx 16213 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
32 | 30, 31 | neeqtrri 2993 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) |
33 | 9, 32 | setsnid 16088 | . . . 4 ⊢ (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
34 | 29, 33 | eqtri 2770 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
35 | 6, 34 | syl6reqr 2801 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
36 | 7 | str0 16084 | . . 3 ⊢ ∅ = (𝐸‘∅) |
37 | fvprc 6334 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = ∅) | |
38 | 37 | adantr 472 | . . 3 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = ∅) |
39 | reldmtng 22614 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
40 | 39 | ovprc1 6835 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
41 | 40 | adantr 472 | . . . . 5 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐺 toNrmGrp 𝑁) = ∅) |
42 | 1, 41 | syl5eq 2794 | . . . 4 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ∅) |
43 | 42 | fveq2d 6344 | . . 3 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑇) = (𝐸‘∅)) |
44 | 36, 38, 43 | 3eqtr4a 2808 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
45 | 35, 44 | pm2.61ian 866 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 Vcvv 3328 ∅c0 4046 〈cop 4315 class class class wbr 4792 ∘ ccom 5258 ‘cfv 6037 (class class class)co 6801 ℝcr 10098 1c1 10100 < clt 10237 ℕcn 11183 2c2 11233 9c9 11240 ;cdc 11656 ndxcnx 16027 sSet csts 16028 Slot cslot 16029 TopSetcts 16120 distcds 16123 -gcsg 17596 MetOpencmopn 19909 toNrmGrp ctng 22555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-7 11247 df-8 11248 df-9 11249 df-n0 11456 df-z 11541 df-dec 11657 df-ndx 16033 df-slot 16034 df-sets 16037 df-tset 16133 df-ds 16137 df-tng 22561 |
This theorem is referenced by: tngbas 22617 tngplusg 22618 tngmulr 22620 tngsca 22621 tngvsca 22622 tngip 22623 |
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