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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoeq1 | Structured version Visualization version GIF version | ||
| Description: Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.) |
| Ref | Expression |
|---|---|
| tendof.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| tendof.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| tendof.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
| Ref | Expression |
|---|---|
| tendoeq1 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp3 1139 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) | |
| 2 | simp1 1137 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 3 | simp2l 1200 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 ∈ 𝐸) | |
| 4 | tendof.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 5 | tendof.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 6 | tendof.e | . . . . . 6 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
| 7 | 4, 5, 6 | tendof 40765 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → 𝑈:𝑇⟶𝑇) |
| 8 | 2, 3, 7 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈:𝑇⟶𝑇) |
| 9 | 8 | ffnd 6737 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 Fn 𝑇) |
| 10 | simp2r 1201 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑉 ∈ 𝐸) | |
| 11 | 4, 5, 6 | tendof 40765 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸) → 𝑉:𝑇⟶𝑇) |
| 12 | 2, 10, 11 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑉:𝑇⟶𝑇) |
| 13 | 12 | ffnd 6737 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑉 Fn 𝑇) |
| 14 | eqfnfv 7051 | . . 3 ⊢ ((𝑈 Fn 𝑇 ∧ 𝑉 Fn 𝑇) → (𝑈 = 𝑉 ↔ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓))) | |
| 15 | 9, 13, 14 | syl2anc 584 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → (𝑈 = 𝑉 ↔ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓))) |
| 16 | 1, 15 | mpbird 257 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 TEndoctendo 40754 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 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-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-tendo 40757 |
| This theorem is referenced by: tendoeq2 40776 tendoplcom 40784 tendoplass 40785 tendodi1 40786 tendodi2 40787 tendo0pl 40793 tendoipl 40799 |
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