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Mirrors > Home > MPE Home > Th. List > tnglemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of tnglem 23996 as of 31-Oct-2024. Lemma for tngbas 23998 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 17069 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 17068 | . . . . . . . 8 ⊢ (𝐸‘ndx) = 𝐾 |
5 | 2 | nnrei 12162 | . . . . . . . 8 ⊢ 𝐾 ∈ ℝ |
6 | 4, 5 | eqeltri 2834 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ ℝ |
7 | tnglemOLD.4 | . . . . . . . . 9 ⊢ 𝐾 < 9 | |
8 | 4, 7 | eqbrtri 5126 | . . . . . . . 8 ⊢ (𝐸‘ndx) < 9 |
9 | 1nn 12164 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
10 | 2nn0 12430 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
11 | 9nn0 12437 | . . . . . . . . 9 ⊢ 9 ∈ ℕ0 | |
12 | 9lt10 12749 | . . . . . . . . 9 ⊢ 9 < ;10 | |
13 | 9, 10, 11, 12 | declti 12656 | . . . . . . . 8 ⊢ 9 < ;12 |
14 | 9re 12252 | . . . . . . . . 9 ⊢ 9 ∈ ℝ | |
15 | 1nn0 12429 | . . . . . . . . . . 11 ⊢ 1 ∈ ℕ0 | |
16 | 15, 10 | deccl 12633 | . . . . . . . . . 10 ⊢ ;12 ∈ ℕ0 |
17 | 16 | nn0rei 12424 | . . . . . . . . 9 ⊢ ;12 ∈ ℝ |
18 | 6, 14, 17 | lttri 11281 | . . . . . . . 8 ⊢ (((𝐸‘ndx) < 9 ∧ 9 < ;12) → (𝐸‘ndx) < ;12) |
19 | 8, 13, 18 | mp2an 690 | . . . . . . 7 ⊢ (𝐸‘ndx) < ;12 |
20 | 6, 19 | ltneii 11268 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ ;12 |
21 | dsndx 17266 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
22 | 20, 21 | neeqtrri 3017 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (dist‘ndx) |
23 | 3, 22 | setsnid 17081 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) |
24 | 6, 8 | ltneii 11268 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 9 |
25 | tsetndx 17233 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
26 | 24, 25 | neeqtrri 3017 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) |
27 | 3, 26 | setsnid 17081 | . . . 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 23995 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
34 | 33 | fveq2d 6846 | . . 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 17061 | . . 3 ⊢ ∅ = (𝐸‘∅) |
37 | fvprc 6834 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = ∅) | |
38 | 37 | adantr 481 | . . 3 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = ∅) |
39 | reldmtng 23994 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
40 | 39 | ovprc1 7396 | . . . . . 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 6846 | . . 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 810 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∅c0 4282 〈cop 4592 class class class wbr 5105 ∘ ccom 5637 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 1c1 11052 < clt 11189 ℕcn 12153 2c2 12208 9c9 12215 ;cdc 12618 sSet csts 17035 Slot cslot 17053 ndxcnx 17065 TopSetcts 17139 distcds 17142 -gcsg 18750 MetOpencmopn 20786 toNrmGrp ctng 23934 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-sets 17036 df-slot 17054 df-ndx 17066 df-tset 17152 df-ds 17155 df-tng 23940 |
This theorem is referenced by: tngbasOLD 23999 tngplusgOLD 24001 tngmulrOLD 24004 tngscaOLD 24006 tngvscaOLD 24008 tngipOLD 24010 |
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