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Mirrors > Home > MPE Home > Th. List > tnglem | Structured version Visualization version GIF version |
Description: Lemma for tngbas 23249 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 2821 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2821 | . . . . 5 ⊢ (𝑁 ∘ (-g‘𝐺)) = (𝑁 ∘ (-g‘𝐺)) | |
4 | eqid 2821 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ (-g‘𝐺))) = (MetOpen‘(𝑁 ∘ (-g‘𝐺))) | |
5 | 1, 2, 3, 4 | tngval 23247 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
6 | 5 | fveq2d 6673 | . . 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 16508 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) |
10 | 7, 8 | ndxarg 16507 | . . . . . . . 8 ⊢ (𝐸‘ndx) = 𝐾 |
11 | 8 | nnrei 11646 | . . . . . . . 8 ⊢ 𝐾 ∈ ℝ |
12 | 10, 11 | eqeltri 2909 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ ℝ |
13 | tnglem.4 | . . . . . . . . 9 ⊢ 𝐾 < 9 | |
14 | 10, 13 | eqbrtri 5086 | . . . . . . . 8 ⊢ (𝐸‘ndx) < 9 |
15 | 1nn 11648 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
16 | 2nn0 11913 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
17 | 9nn0 11920 | . . . . . . . . 9 ⊢ 9 ∈ ℕ0 | |
18 | 9lt10 12228 | . . . . . . . . 9 ⊢ 9 < ;10 | |
19 | 15, 16, 17, 18 | declti 12135 | . . . . . . . 8 ⊢ 9 < ;12 |
20 | 9re 11735 | . . . . . . . . 9 ⊢ 9 ∈ ℝ | |
21 | 1nn0 11912 | . . . . . . . . . . 11 ⊢ 1 ∈ ℕ0 | |
22 | 21, 16 | deccl 12112 | . . . . . . . . . 10 ⊢ ;12 ∈ ℕ0 |
23 | 22 | nn0rei 11907 | . . . . . . . . 9 ⊢ ;12 ∈ ℝ |
24 | 12, 20, 23 | lttri 10765 | . . . . . . . 8 ⊢ (((𝐸‘ndx) < 9 ∧ 9 < ;12) → (𝐸‘ndx) < ;12) |
25 | 14, 19, 24 | mp2an 690 | . . . . . . 7 ⊢ (𝐸‘ndx) < ;12 |
26 | 12, 25 | ltneii 10752 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ ;12 |
27 | dsndx 16674 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
28 | 26, 27 | neeqtrri 3089 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (dist‘ndx) |
29 | 9, 28 | setsnid 16538 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) |
30 | 12, 14 | ltneii 10752 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 9 |
31 | tsetndx 16658 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
32 | 30, 31 | neeqtrri 3089 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) |
33 | 9, 32 | setsnid 16538 | . . . 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 2844 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
35 | 6, 34 | syl6reqr 2875 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
36 | 7 | str0 16534 | . . 3 ⊢ ∅ = (𝐸‘∅) |
37 | fvprc 6662 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = ∅) | |
38 | 37 | adantr 483 | . . 3 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = ∅) |
39 | reldmtng 23246 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
40 | 39 | ovprc1 7194 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
41 | 40 | adantr 483 | . . . . 5 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐺 toNrmGrp 𝑁) = ∅) |
42 | 1, 41 | syl5eq 2868 | . . . 4 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ∅) |
43 | 42 | fveq2d 6673 | . . 3 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑇) = (𝐸‘∅)) |
44 | 36, 38, 43 | 3eqtr4a 2882 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
45 | 35, 44 | pm2.61ian 810 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 〈cop 4572 class class class wbr 5065 ∘ ccom 5558 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 1c1 10537 < clt 10674 ℕcn 11637 2c2 11691 9c9 11698 ;cdc 12097 ndxcnx 16479 sSet csts 16480 Slot cslot 16481 TopSetcts 16570 distcds 16573 -gcsg 18104 MetOpencmopn 20534 toNrmGrp ctng 23187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-ndx 16485 df-slot 16486 df-sets 16489 df-tset 16583 df-ds 16586 df-tng 23193 |
This theorem is referenced by: tngbas 23249 tngplusg 23250 tngmulr 23252 tngsca 23253 tngvsca 23254 tngip 23255 |
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