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Theorem tendoeq2 40757
Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 40807, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)
Hypotheses
Ref Expression
tendoeq2.b 𝐵 = (Base‘𝐾)
tendoeq2.h 𝐻 = (LHyp‘𝐾)
tendoeq2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoeq2.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
tendoeq2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) → 𝑈 = 𝑉)
Distinct variable groups:   𝑓,𝐸   𝑓,𝐻   𝑓,𝐾   𝑇,𝑓   𝑓,𝑊   𝑈,𝑓   𝑓,𝑉
Allowed substitution hint:   𝐵(𝑓)

Proof of Theorem tendoeq2
StepHypRef Expression
1 tendoeq2.b . . . . . . . 8 𝐵 = (Base‘𝐾)
2 tendoeq2.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
3 tendoeq2.e . . . . . . . 8 𝐸 = ((TEndo‘𝐾)‘𝑊)
41, 2, 3tendoid 40756 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
54adantrr 717 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
61, 2, 3tendoid 40756 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑉𝐸) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
76adantrl 716 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
85, 7eqtr4d 2778 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵)))
9 fveq2 6907 . . . . . 6 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑈‘( I ↾ 𝐵)))
10 fveq2 6907 . . . . . 6 (𝑓 = ( I ↾ 𝐵) → (𝑉𝑓) = (𝑉‘( I ↾ 𝐵)))
119, 10eqeq12d 2751 . . . . 5 (𝑓 = ( I ↾ 𝐵) → ((𝑈𝑓) = (𝑉𝑓) ↔ (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵))))
128, 11syl5ibrcom 247 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)))
1312ralrimivw 3148 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → ∀𝑓𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)))
14 r19.26 3109 . . . . 5 (∀𝑓𝑇 ((𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) ↔ (∀𝑓𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ ∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))))
15 jaob 963 . . . . . . 7 (((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (𝑈𝑓) = (𝑉𝑓)) ↔ ((𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))))
16 exmidne 2948 . . . . . . . 8 (𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵))
17 pm5.5 361 . . . . . . . 8 ((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (𝑈𝑓) = (𝑉𝑓)) ↔ (𝑈𝑓) = (𝑉𝑓)))
1816, 17ax-mp 5 . . . . . . 7 (((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (𝑈𝑓) = (𝑉𝑓)) ↔ (𝑈𝑓) = (𝑉𝑓))
1915, 18bitr3i 277 . . . . . 6 (((𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) ↔ (𝑈𝑓) = (𝑉𝑓))
2019ralbii 3091 . . . . 5 (∀𝑓𝑇 ((𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) ↔ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓))
2114, 20bitr3i 277 . . . 4 ((∀𝑓𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ ∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) ↔ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓))
22 tendoeq2.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
232, 22, 3tendoeq1 40747 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑈 = 𝑉)
24233expia 1120 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓) → 𝑈 = 𝑉))
2521, 24biimtrid 242 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → ((∀𝑓𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ ∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) → 𝑈 = 𝑉))
2613, 25mpand 695 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) → 𝑈 = 𝑉))
27263impia 1116 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) → 𝑈 = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059   I cid 5582  cres 5691  cfv 6563  Basecbs 17245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-tendo 40738
This theorem is referenced by:  tendocan  40807
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