Theorem tendoeq2 41403

Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 41453, 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 41402	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
5	4	adantrr 727	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
6	1, 2, 3	tendoid 41402	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
7	6	adantrl 726	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
8	5, 7	eqtr4d 2802	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵)))
9		fveq2 6869	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑈‘( I ↾ 𝐵)))
10		fveq2 6869	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑉‘𝑓) = (𝑉‘( I ↾ 𝐵)))
11	9, 10	eqeq12d 2780	. . . . 5 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑈‘𝑓) = (𝑉‘𝑓) ↔ (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵))))
12	8, 11	syl5ibrcom 249	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
13	12	ralrimivw 3160	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → ∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
14		r19.26 3124	. . . . 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 974	. . . . . . 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 2969	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵))
17		pm5.5 363	. . . . . . . 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 279	. . . . . 6 ⊢ (((𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) ↔ (𝑈‘𝑓) = (𝑉‘𝑓))
20	19	ralbii 3110	. . . . 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 279	. . . 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 41393	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉)
24	23	3expia 1135	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓) → 𝑈 = 𝑉))
25	21, 24	biimtrid 244	. . 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 705	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉))
27	26	3impia 1131	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 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078   I cid 5543   ↾ cres 5651  ‘cfv 6523  Basecbs 17247  HLchlt 39979  LHypclh 40613  LTrncltrn 40730  TEndoctendo 41381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-map 8812  df-proset 18328  df-poset 18347  df-plt 18362  df-lub 18378  df-glb 18379  df-join 18380  df-meet 18381  df-p0 18457  df-p1 18458  df-lat 18466  df-clat 18533  df-oposet 39805  df-ol 39807  df-oml 39808  df-covers 39895  df-ats 39896  df-atl 39927  df-cvlat 39951  df-hlat 39980  df-lhyp 40617  df-laut 40618  df-ldil 40733  df-ltrn 40734  df-trl 40788  df-tendo 41384

This theorem is referenced by:  tendocan  41453