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Theorem ltrncoidN 37144
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 1183 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1185 . . . . . . . 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 37140 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺:𝐵1-1-onto𝐵)
71, 2, 6syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺:𝐵1-1-onto𝐵)
8 f1ococnv1 6636 . . . . . . 7 (𝐺:𝐵1-1-onto𝐵 → (𝐺𝐺) = ( I ↾ 𝐵))
97, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐺𝐺) = ( I ↾ 𝐵))
109coeq2d 5726 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
11 simpl2 1184 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹𝑇)
123, 4, 5ltrn1o 37140 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:𝐵1-1-onto𝐵)
131, 11, 12syl2anc 584 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐵1-1-onto𝐵)
14 f1of 6608 . . . . . 6 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
15 fcoi1 6545 . . . . . 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 2854 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = (𝐹 ∘ (𝐺𝐺)))
18 coass 6111 . . . 4 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
1917, 18syl6eqr 2871 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = ((𝐹𝐺) ∘ 𝐺))
20 simpr 485 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → (𝐹𝐺) = ( I ↾ 𝐵))
2120coeq1d 5725 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → ((𝐹𝐺) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22 f1of 6608 . . . . 5 (𝐺:𝐵1-1-onto𝐵𝐺:𝐵𝐵)
23 fcoi2 6546 . . . . 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 2853 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → ((𝐹𝐺) ∘ 𝐺) = 𝐺)
2619, 25eqtrd 2853 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
27 simpr 485 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
2827coeq1d 5725 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐹𝐺) = (𝐺𝐺))
29 simpl1 1183 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐾 ∈ HL ∧ 𝑊𝐻))
30 simpl3 1185 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐺𝑇)
3129, 30, 6syl2anc 584 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → 𝐺:𝐵1-1-onto𝐵)
32 f1ococnv2 6634 . . . 4 (𝐺:𝐵1-1-onto𝐵 → (𝐺𝐺) = ( I ↾ 𝐵))
3331, 32syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐺𝐺) = ( I ↾ 𝐵))
3428, 33eqtrd 2853 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝐹 = 𝐺) → (𝐹𝐺) = ( I ↾ 𝐵))
3526, 34impbida 797 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → ((𝐹𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105   I cid 5452  ccnv 5547  cres 5550  ccom 5552  wf 6344  1-1-ontowf1o 6347  cfv 6348  Basecbs 16471  HLchlt 36366  LHypclh 37000  LTrncltrn 37117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-laut 37005  df-ldil 37120  df-ltrn 37121
This theorem is referenced by:  tendospcanN  38039
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