Theorem tendoeq2 40798

Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 40848, 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) → 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

Step	Hyp	Ref	Expression
1		tendoeq2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
2		tendoeq2.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendoeq2.e	. . . . . . . 8 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4	1, 2, 3	tendoid 40797	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
5	4	adantrr 717	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
6	1, 2, 3	tendoid 40797	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
7	6	adantrl 716	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
8	5, 7	eqtr4d 2774	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵)))
9		fveq2 6881	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑈‘( I ↾ 𝐵)))
10		fveq2 6881	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑉‘𝑓) = (𝑉‘( I ↾ 𝐵)))
11	9, 10	eqeq12d 2752	. . . . 5 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑈‘𝑓) = (𝑉‘𝑓) ↔ (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵))))
12	8, 11	syl5ibrcom 247	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
13	12	ralrimivw 3137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → ∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
14		r19.26 3099	. . . . 5 ⊢ (∀𝑓 ∈ 𝑇 ((𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) ↔ (∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ ∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))))
15		jaob 963	. . . . . . 7 ⊢ (((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (𝑈‘𝑓) = (𝑉‘𝑓)) ↔ ((𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))))
16		exmidne 2943	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵))
17		pm5.5 361	. . . . . . . 8 ⊢ ((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (𝑈‘𝑓) = (𝑉‘𝑓)) ↔ (𝑈‘𝑓) = (𝑉‘𝑓)))
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (((𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵)) → (𝑈‘𝑓) = (𝑉‘𝑓)) ↔ (𝑈‘𝑓) = (𝑉‘𝑓))
19	15, 18	bitr3i 277	. . . . . 6 ⊢ (((𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) ↔ (𝑈‘𝑓) = (𝑉‘𝑓))
20	19	ralbii 3083	. . . . 5 ⊢ (∀𝑓 ∈ 𝑇 ((𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) ↔ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓))
21	14, 20	bitr3i 277	. . . 4 ⊢ ((∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ ∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) ↔ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓))
22		tendoeq2.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
23	2, 22, 3	tendoeq1 40788	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉)
24	23	3expia 1121	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓) → 𝑈 = 𝑉))
25	21, 24	biimtrid 242	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → ((∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ ∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) → 𝑈 = 𝑉))
26	13, 25	mpand 695	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉))
27	26	3impia 1117	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) → 𝑈 = 𝑉)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052   I cid 5552   ↾ cres 5661  ‘cfv 6536  Basecbs 17233  HLchlt 39373  LHypclh 40008  LTrncltrn 40125  TEndoctendo 40776

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-proset 18311  df-poset 18330  df-plt 18345  df-lub 18361  df-glb 18362  df-join 18363  df-meet 18364  df-p0 18440  df-p1 18441  df-lat 18447  df-clat 18514  df-oposet 39199  df-ol 39201  df-oml 39202  df-covers 39289  df-ats 39290  df-atl 39321  df-cvlat 39345  df-hlat 39374  df-lhyp 40012  df-laut 40013  df-ldil 40128  df-ltrn 40129  df-trl 40183  df-tendo 40779

This theorem is referenced by:  tendocan  40848