Theorem ltrncoidN 40827

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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))

Proof of Theorem ltrncoidN

Step	Hyp	Ref	Expression
1		simpl1 1208	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1210	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺 ∈ 𝑇)
3		ltrn1o.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
4		ltrn1o.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
5		ltrn1o.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6	3, 4, 5	ltrn1o 40823	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:𝐵–1-1-onto→𝐵)
7	1, 2, 6	syl2anc 595	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺:𝐵–1-1-onto→𝐵)
8		f1ococnv1 6851	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
9	7, 8	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
10	9	coeq2d 5849	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ (◡𝐺 ∘ 𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
11		simpl2 1209	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 ∈ 𝑇)
12	3, 4, 5	ltrn1o 40823	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
13	1, 11, 12	syl2anc 595	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐵–1-1-onto→𝐵)
14		f1of 6821	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
15		fcoi1 6753	. . . . . 6 ⊢ (𝐹:𝐵⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
16	13, 14, 15	3syl 19	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
17	10, 16	eqtr2d 2805	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = (𝐹 ∘ (◡𝐺 ∘ 𝐺)))
18		coass 6268	. . . 4 ⊢ ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (𝐹 ∘ (◡𝐺 ∘ 𝐺))
19	17, 18	eqtr4di 2822	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = ((𝐹 ∘ ◡𝐺) ∘ 𝐺))
20		simpr 489	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))
21	20	coeq1d 5848	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22		f1of 6821	. . . . 5 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵⟶𝐵)
23		fcoi2 6754	. . . . 5 ⊢ (𝐺:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
24	7, 22, 23	3syl 19	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
25	21, 24	eqtrd 2804	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = 𝐺)
26	19, 25	eqtrd 2804	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
27		simpr 489	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
28	27	coeq1d 5848	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐺))
29		simpl1 1208	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30		simpl3 1210	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺 ∈ 𝑇)
31	29, 30, 6	syl2anc 595	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺:𝐵–1-1-onto→𝐵)
32		f1ococnv2 6849	. . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝐵))
33	31, 32	syl 18	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝐵))
34	28, 33	eqtrd 2804	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))
35	26, 34	impbida 812	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   I cid 5556  ^◡ccnv 5661   ↾ cres 5664   ∘ ccom 5666  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  Basecbs 17269  HLchlt 40049  LHypclh 40683  LTrncltrn 40800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826  df-laut 40688  df-ldil 40803  df-ltrn 40804

This theorem is referenced by:  tendospcanN  41722