Theorem tendoeq2 40731

Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 40781, 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 40730	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
5	4	adantrr 716	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
6	1, 2, 3	tendoid 40730	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
7	6	adantrl 715	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
8	5, 7	eqtr4d 2783	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵)))
9		fveq2 6920	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑈‘( I ↾ 𝐵)))
10		fveq2 6920	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑉‘𝑓) = (𝑉‘( I ↾ 𝐵)))
11	9, 10	eqeq12d 2756	. . . . 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 3156	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → ∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
14		r19.26 3117	. . . . 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 962	. . . . . . 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 2956	. . . . . . . 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 3099	. . . . 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 40721	. . . . 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 694	. 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 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067   I cid 5592   ↾ cres 5702  ‘cfv 6573  Basecbs 17258  HLchlt 39306  LHypclh 39941  LTrncltrn 40058  TEndoctendo 40709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-p1 18496  df-lat 18502  df-clat 18569  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-lhyp 39945  df-laut 39946  df-ldil 40061  df-ltrn 40062  df-trl 40116  df-tendo 40712

This theorem is referenced by:  tendocan  40781