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Theorem tendoeq1 36572
 Description: Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendof.h 𝐻 = (LHyp‘𝐾)
tendof.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendof.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
tendoeq1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑈 = 𝑉)
Distinct variable groups:   𝑓,𝐾   𝑇,𝑓   𝑓,𝑊   𝑈,𝑓   𝑓,𝑉
Allowed substitution hints:   𝐸(𝑓)   𝐻(𝑓)

Proof of Theorem tendoeq1
StepHypRef Expression
1 simp3 1133 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓))
2 simp1 1131 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simp2l 1242 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑈𝐸)
4 tendof.h . . . . . 6 𝐻 = (LHyp‘𝐾)
5 tendof.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 tendof.e . . . . . 6 𝐸 = ((TEndo‘𝐾)‘𝑊)
74, 5, 6tendof 36571 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → 𝑈:𝑇𝑇)
82, 3, 7syl2anc 696 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑈:𝑇𝑇)
9 ffn 6206 . . . 4 (𝑈:𝑇𝑇𝑈 Fn 𝑇)
108, 9syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑈 Fn 𝑇)
11 simp2r 1243 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑉𝐸)
124, 5, 6tendof 36571 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑉𝐸) → 𝑉:𝑇𝑇)
132, 11, 12syl2anc 696 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑉:𝑇𝑇)
14 ffn 6206 . . . 4 (𝑉:𝑇𝑇𝑉 Fn 𝑇)
1513, 14syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑉 Fn 𝑇)
16 eqfnfv 6475 . . 3 ((𝑈 Fn 𝑇𝑉 Fn 𝑇) → (𝑈 = 𝑉 ↔ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)))
1710, 15, 16syl2anc 696 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → (𝑈 = 𝑉 ↔ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)))
181, 17mpbird 247 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑈 = 𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  HLchlt 35158  LHypclh 35791  LTrncltrn 35908  TEndoctendo 36560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-tendo 36563 This theorem is referenced by:  tendoeq2  36582  tendoplcom  36590  tendoplass  36591  tendodi1  36592  tendodi2  36593  tendo0pl  36599  tendoipl  36605
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