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Theorem tendoeq2 36549
Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 36599, 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 36548 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
54adantrr 699 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
61, 2, 3tendoid 36548 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑉𝐸) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
76adantrl 698 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
85, 7eqtr4d 2843 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵)))
9 fveq2 6404 . . . . . 6 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑈‘( I ↾ 𝐵)))
10 fveq2 6404 . . . . . 6 (𝑓 = ( I ↾ 𝐵) → (𝑉𝑓) = (𝑉‘( I ↾ 𝐵)))
119, 10eqeq12d 2821 . . . . 5 (𝑓 = ( I ↾ 𝐵) → ((𝑈𝑓) = (𝑉𝑓) ↔ (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵))))
128, 11syl5ibrcom 238 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)))
1312ralrimivw 3155 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → ∀𝑓𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)))
14 r19.26 3252 . . . . 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 975 . . . . . . 7 (((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (𝑈𝑓) = (𝑉𝑓)) ↔ ((𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))))
16 exmidne 2988 . . . . . . . 8 (𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵))
17 pm5.5 352 . . . . . . . 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 268 . . . . . 6 (((𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) ↔ (𝑈𝑓) = (𝑉𝑓))
2019ralbii 3168 . . . . 5 (∀𝑓𝑇 ((𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) ↔ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓))
2114, 20bitr3i 268 . . . 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 36539 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓)) → 𝑈 = 𝑉)
24233expia 1143 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (∀𝑓𝑇 (𝑈𝑓) = (𝑉𝑓) → 𝑈 = 𝑉))
2521, 24syl5bi 233 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → ((∀𝑓𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) ∧ ∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) → 𝑈 = 𝑉))
2613, 25mpand 678 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸)) → (∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓)) → 𝑈 = 𝑉))
27263impia 1138 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ ∀𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈𝑓) = (𝑉𝑓))) → 𝑈 = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wral 3096   I cid 5218  cres 5313  cfv 6097  Basecbs 16064  HLchlt 35125  LHypclh 35759  LTrncltrn 35876  TEndoctendo 36527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-map 8090  df-proset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-p1 17241  df-lat 17247  df-clat 17309  df-oposet 34951  df-ol 34953  df-oml 34954  df-covers 35041  df-ats 35042  df-atl 35073  df-cvlat 35097  df-hlat 35126  df-lhyp 35763  df-laut 35764  df-ldil 35879  df-ltrn 35880  df-trl 35934  df-tendo 36530
This theorem is referenced by:  tendocan  36599
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