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| Mirrors > Home > MPE Home > Th. List > tnglem | Structured version Visualization version GIF version | ||
| Description: Lemma for tngbas 24655 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 17245 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ (-g‘𝐺))〉)) |
| 4 | tnglem.t | . . . . 5 ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx) | |
| 5 | 1, 4 | setsnid 17245 | . . . 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 2765 | . . 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 2737 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (𝑁 ∘ (-g‘𝐺)) = (𝑁 ∘ (-g‘𝐺)) | |
| 10 | eqid 2737 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ (-g‘𝐺))) = (MetOpen‘(𝑁 ∘ (-g‘𝐺))) | |
| 11 | 7, 8, 9, 10 | tngval 24652 | . . . 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 2796 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝐺) = (𝐸‘𝑇)) |
| 14 | 1 | str0 17226 | . . . . 5 ⊢ ∅ = (𝐸‘∅) |
| 15 | 14 | eqcomi 2746 | . . . 4 ⊢ (𝐸‘∅) = ∅ |
| 16 | reldmtng 24651 | . . . 4 ⊢ Rel dom toNrmGrp | |
| 17 | 15, 7, 16 | oveqprc 17229 | . . 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 〈cop 4632 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 TopSetcts 17303 distcds 17306 -gcsg 18953 MetOpencmopn 21354 toNrmGrp ctng 24591 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 df-slot 17219 df-tng 24597 |
| This theorem is referenced by: tngbas 24655 tngplusg 24657 tngmulr 24660 tngsca 24662 tngvsca 24664 tngip 24666 |
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