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Mirrors > Home > MPE Home > Th. List > tnglem | Structured version Visualization version GIF version |
Description: Lemma for tngbas 24670 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 17242 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) |
4 | tnglem.t | . . . . 5 ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) | |
5 | 1, 4 | setsnid 17242 | . . . 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 2762 | . . 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 2734 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
9 | eqid 2734 | . . . . 5 ⊢ (𝑁 ∘ (-g‘𝐺)) = (𝑁 ∘ (-g‘𝐺)) | |
10 | eqid 2734 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ (-g‘𝐺))) = (MetOpen‘(𝑁 ∘ (-g‘𝐺))) | |
11 | 7, 8, 9, 10 | tngval 24667 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g‘𝐺)))〉)) |
12 | 11 | fveq2d 6910 | . . 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 2793 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
14 | 1 | str0 17222 | . . . . 5 ⊢ ∅ = (𝐸‘∅) |
15 | 14 | eqcomi 2743 | . . . 4 ⊢ (𝐸‘∅) = ∅ |
16 | reldmtng 24666 | . . . 4 ⊢ Rel dom toNrmGrp | |
17 | 15, 7, 16 | oveqprc 17225 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑇)) |
18 | 17 | adantr 480 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
19 | 13, 18 | pm2.61ian 812 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∅c0 4338 〈cop 4636 ∘ ccom 5692 ‘cfv 6562 (class class class)co 7430 sSet csts 17196 Slot cslot 17214 ndxcnx 17226 TopSetcts 17303 distcds 17306 -gcsg 18965 MetOpencmopn 21371 toNrmGrp ctng 24606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-res 5700 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-sets 17197 df-slot 17215 df-tng 24612 |
This theorem is referenced by: tngbas 24670 tngplusg 24672 tngmulr 24675 tngsca 24677 tngvsca 24679 tngip 24681 |
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