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Mirrors > Home > MPE Home > Th. List > tnglem | Structured version Visualization version GIF version |
Description: Lemma for tngbas 24142 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.) |
Ref | Expression |
---|---|
tngbas.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tnglem.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
tnglem.t | ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) |
tnglem.d | ⊢ (𝐸‘ndx) ≠ (dist‘ndx) |
Ref | Expression |
---|---|
tnglem | ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tnglem.e | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | tnglem.d | . . . . 5 ⊢ (𝐸‘ndx) ≠ (dist‘ndx) | |
3 | 1, 2 | setsnid 17138 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) |
4 | tnglem.t | . . . . 5 ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) | |
5 | 1, 4 | setsnid 17138 | . . . 4 ⊢ (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
6 | 3, 5 | eqtri 2760 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
7 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
8 | eqid 2732 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
9 | eqid 2732 | . . . . 5 ⊢ (𝑁 ∘ (-g‘𝐺)) = (𝑁 ∘ (-g‘𝐺)) | |
10 | eqid 2732 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ (-g‘𝐺))) = (MetOpen‘(𝑁 ∘ (-g‘𝐺))) | |
11 | 7, 8, 9, 10 | tngval 24139 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
12 | 11 | fveq2d 6892 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑇) = (𝐸‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉))) |
13 | 6, 12 | eqtr4id 2791 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
14 | 1 | str0 17118 | . . . . 5 ⊢ ∅ = (𝐸‘∅) |
15 | 14 | eqcomi 2741 | . . . 4 ⊢ (𝐸‘∅) = ∅ |
16 | reldmtng 24138 | . . . 4 ⊢ Rel dom toNrmGrp | |
17 | 15, 7, 16 | oveqprc 17121 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑇)) |
18 | 17 | adantr 481 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
19 | 13, 18 | 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 ≠ wne 2940 Vcvv 3474 ∅c0 4321 〈cop 4633 ∘ ccom 5679 ‘cfv 6540 (class class class)co 7405 sSet csts 17092 Slot cslot 17110 ndxcnx 17122 TopSetcts 17199 distcds 17202 -gcsg 18817 MetOpencmopn 20926 toNrmGrp ctng 24078 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-res 5687 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-sets 17093 df-slot 17111 df-tng 24084 |
This theorem is referenced by: tngbas 24142 tngplusg 24144 tngmulr 24147 tngsca 24149 tngvsca 24151 tngip 24153 |
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