Theorem tendoeq2 38715

Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 38765, 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 38714	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
5	4	adantrr 713	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
6	1, 2, 3	tendoid 38714	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
7	6	adantrl 712	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
8	5, 7	eqtr4d 2781	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵)))
9		fveq2 6756	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑈‘( I ↾ 𝐵)))
10		fveq2 6756	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑉‘𝑓) = (𝑉‘( I ↾ 𝐵)))
11	9, 10	eqeq12d 2754	. . . . 5 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑈‘𝑓) = (𝑉‘𝑓) ↔ (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵))))
12	8, 11	syl5ibrcom 246	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
13	12	ralrimivw 3108	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → ∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
14		r19.26 3094	. . . . 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 958	. . . . . . 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 2952	. . . . . . . 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 276	. . . . . 6 ⊢ (((𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) ↔ (𝑈‘𝑓) = (𝑉‘𝑓))
20	19	ralbii 3090	. . . . 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 276	. . . 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 38705	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉)
24	23	3expia 1119	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓) → 𝑈 = 𝑉))
25	21, 24	syl5bi 241	. . 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 691	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉))
27	26	3impia 1115	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 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   I cid 5479   ↾ cres 5582  ‘cfv 6418  Basecbs 16840  HLchlt 37291  LHypclh 37925  LTrncltrn 38042  TEndoctendo 38693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tendo 38696

This theorem is referenced by:  tendocan  38765