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Theorem ltrncoidN 36202
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 1248 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1252 . . . . . . . 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 36198 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺:𝐵1-1-onto𝐵)
71, 2, 6syl2anc 581 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺:𝐵1-1-onto𝐵)
8 f1ococnv1 6405 . . . . . . 7 (𝐺:𝐵1-1-onto𝐵 → (𝐺𝐺) = ( I ↾ 𝐵))
97, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐺𝐺) = ( I ↾ 𝐵))
109coeq2d 5516 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
11 simpl2 1250 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹𝑇)
123, 4, 5ltrn1o 36198 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:𝐵1-1-onto𝐵)
131, 11, 12syl2anc 581 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐵1-1-onto𝐵)
14 f1of 6377 . . . . . 6 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
15 fcoi1 6314 . . . . . 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 2861 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = (𝐹 ∘ (𝐺𝐺)))
18 coass 5894 . . . 4 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
1917, 18syl6eqr 2878 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = ((𝐹𝐺) ∘ 𝐺))
20 simpr 479 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐹𝐺) = ( I ↾ 𝐵))
2120coeq1d 5515 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → ((𝐹𝐺) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22 f1of 6377 . . . . 5 (𝐺:𝐵1-1-onto𝐵𝐺:𝐵𝐵)
23 fcoi2 6315 . . . . 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 2860 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → ((𝐹𝐺) ∘ 𝐺) = 𝐺)
2619, 25eqtrd 2860 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
27 simpr 479 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
2827coeq1d 5515 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐹𝐺) = (𝐺𝐺))
29 simpl1 1248 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐾 ∈ HL ∧ 𝑊𝐻))
30 simpl3 1252 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐺𝑇)
3129, 30, 6syl2anc 581 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐺:𝐵1-1-onto𝐵)
32 f1ococnv2 6403 . . . 4 (𝐺:𝐵1-1-onto𝐵 → (𝐺𝐺) = ( I ↾ 𝐵))
3331, 32syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐺𝐺) = ( I ↾ 𝐵))
3428, 33eqtrd 2860 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐹𝐺) = ( I ↾ 𝐵))
3526, 34impbida 837 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → ((𝐹𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166   I cid 5248  ccnv 5340  cres 5343  ccom 5345  wf 6118  1-1-ontowf1o 6121  cfv 6122  Basecbs 16221  HLchlt 35424  LHypclh 36058  LTrncltrn 36175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-map 8123  df-laut 36063  df-ldil 36178  df-ltrn 36179
This theorem is referenced by:  tendospcanN  37097
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