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Theorem ltrncoidN 35929
 Description: Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analogue of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrn1o.b 𝐵 = (Base‘𝐾)
ltrn1o.h 𝐻 = (LHyp‘𝐾)
ltrn1o.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrncoidN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → ((𝐹𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))

Proof of Theorem ltrncoidN
StepHypRef Expression
1 simpl1 1226 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1230 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺𝑇)
3 ltrn1o.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
4 ltrn1o.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
5 ltrn1o.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
63, 4, 5ltrn1o 35925 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺:𝐵1-1-onto𝐵)
71, 2, 6syl2anc 565 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺:𝐵1-1-onto𝐵)
8 f1ococnv1 6306 . . . . . . 7 (𝐺:𝐵1-1-onto𝐵 → (𝐺𝐺) = ( I ↾ 𝐵))
97, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐺𝐺) = ( I ↾ 𝐵))
109coeq2d 5423 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
11 simpl2 1228 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹𝑇)
123, 4, 5ltrn1o 35925 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:𝐵1-1-onto𝐵)
131, 11, 12syl2anc 565 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐵1-1-onto𝐵)
14 f1of 6278 . . . . . 6 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
15 fcoi1 6218 . . . . . 6 (𝐹:𝐵𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
1613, 14, 153syl 18 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
1710, 16eqtr2d 2805 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = (𝐹 ∘ (𝐺𝐺)))
18 coass 5798 . . . 4 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
1917, 18syl6eqr 2822 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = ((𝐹𝐺) ∘ 𝐺))
20 simpr 471 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐹𝐺) = ( I ↾ 𝐵))
2120coeq1d 5422 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → ((𝐹𝐺) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22 f1of 6278 . . . . 5 (𝐺:𝐵1-1-onto𝐵𝐺:𝐵𝐵)
23 fcoi2 6219 . . . . 5 (𝐺:𝐵𝐵 → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
247, 22, 233syl 18 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
2521, 24eqtrd 2804 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → ((𝐹𝐺) ∘ 𝐺) = 𝐺)
2619, 25eqtrd 2804 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
27 simpr 471 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
2827coeq1d 5422 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐹𝐺) = (𝐺𝐺))
29 simpl1 1226 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐾 ∈ HL ∧ 𝑊𝐻))
30 simpl3 1230 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐺𝑇)
3129, 30, 6syl2anc 565 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐺:𝐵1-1-onto𝐵)
32 f1ococnv2 6304 . . . 4 (𝐺:𝐵1-1-onto𝐵 → (𝐺𝐺) = ( I ↾ 𝐵))
3331, 32syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐺𝐺) = ( I ↾ 𝐵))
3428, 33eqtrd 2804 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐹𝐺) = ( I ↾ 𝐵))
3526, 34impbida 794 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → ((𝐹𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144   I cid 5156  ◡ccnv 5248   ↾ cres 5251   ∘ ccom 5253  ⟶wf 6027  –1-1-onto→wf1o 6030  ‘cfv 6031  Basecbs 16063  HLchlt 35152  LHypclh 35785  LTrncltrn 35902 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-laut 35790  df-ldil 35905  df-ltrn 35906 This theorem is referenced by:  tendospcanN  36826
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