Theorem ltrncoidN 40635

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 1199	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1201	. . . . . . . 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 40631	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:𝐵–1-1-onto→𝐵)
7	1, 2, 6	syl2anc 591	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺:𝐵–1-1-onto→𝐵)
8		f1ococnv1 6800	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
9	7, 8	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
10	9	coeq2d 5807	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ (◡𝐺 ∘ 𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
11		simpl2 1200	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 ∈ 𝑇)
12	3, 4, 5	ltrn1o 40631	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
13	1, 11, 12	syl2anc 591	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐵–1-1-onto→𝐵)
14		f1of 6771	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
15		fcoi1 6705	. . . . . 6 ⊢ (𝐹:𝐵⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
16	13, 14, 15	3syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
17	10, 16	eqtr2d 2777	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = (𝐹 ∘ (◡𝐺 ∘ 𝐺)))
18		coass 6221	. . . 4 ⊢ ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (𝐹 ∘ (◡𝐺 ∘ 𝐺))
19	17, 18	eqtr4di 2794	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = ((𝐹 ∘ ◡𝐺) ∘ 𝐺))
20		simpr 486	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))
21	20	coeq1d 5806	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22		f1of 6771	. . . . 5 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵⟶𝐵)
23		fcoi2 6706	. . . . 5 ⊢ (𝐺:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
24	7, 22, 23	3syl 18	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
25	21, 24	eqtrd 2776	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = 𝐺)
26	19, 25	eqtrd 2776	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
27		simpr 486	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
28	27	coeq1d 5806	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐺))
29		simpl1 1199	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30		simpl3 1201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺 ∈ 𝑇)
31	29, 30, 6	syl2anc 591	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺:𝐵–1-1-onto→𝐵)
32		f1ococnv2 6798	. . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝐵))
33	31, 32	syl 17	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝐵))
34	28, 33	eqtrd 2776	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))
35	26, 34	impbida 807	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   I cid 5515  ^◡ccnv 5620   ↾ cres 5623   ∘ ccom 5625  ⟶wf 6485  –1-1-onto→wf1o 6488  ‘cfv 6489  Basecbs 17174  HLchlt 39857  LHypclh 40491  LTrncltrn 40608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-laut 40496  df-ldil 40611  df-ltrn 40612

This theorem is referenced by:  tendospcanN  41530