Theorem tendoeq2 41832

Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 41882, 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 41831	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
5	4	adantrr 730	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
6	1, 2, 3	tendoid 41831	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
7	6	adantrl 729	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑉‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
8	5, 7	eqtr4d 2798	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵)))
9		fveq2 6885	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑈‘( I ↾ 𝐵)))
10		fveq2 6885	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐵) → (𝑉‘𝑓) = (𝑉‘( I ↾ 𝐵)))
11	9, 10	eqeq12d 2776	. . . . 5 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑈‘𝑓) = (𝑉‘𝑓) ↔ (𝑈‘( I ↾ 𝐵)) = (𝑉‘( I ↾ 𝐵))))
12	8, 11	syl5ibrcom 250	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
13	12	ralrimivw 3158	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → ∀𝑓 ∈ 𝑇 (𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)))
14		r19.26 3122	. . . . 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 976	. . . . . . 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 2965	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ 𝐵) ∨ 𝑓 ≠ ( I ↾ 𝐵))
17		pm5.5 364	. . . . . . . 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 280	. . . . . 6 ⊢ (((𝑓 = ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) ∧ (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) ↔ (𝑈‘𝑓) = (𝑉‘𝑓))
20	19	ralbii 3108	. . . . 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 280	. . . 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 41822	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉)
24	23	3expia 1139	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓) → 𝑈 = 𝑉))
25	21, 24	biimtrid 245	. . 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 708	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉))
27	26	3impia 1135	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) → 𝑈 = 𝑉)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   I cid 5544   ↾ cres 5652  ‘cfv 6538  Basecbs 17387  HLchlt 40408  LHypclh 41042  LTrncltrn 41159  TEndoctendo 41810

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40234  df-ol 40236  df-oml 40237  df-covers 40324  df-ats 40325  df-atl 40356  df-cvlat 40380  df-hlat 40409  df-lhyp 41046  df-laut 41047  df-ldil 41162  df-ltrn 41163  df-trl 41217  df-tendo 41813

This theorem is used by:  tendocan  41882