Theorem ltrncoidN 38069

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 1189	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1191	. . . . . . . 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 38065	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:𝐵–1-1-onto→𝐵)
7	1, 2, 6	syl2anc 583	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺:𝐵–1-1-onto→𝐵)
8		f1ococnv1 6728	. . . . . . 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 5760	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ (◡𝐺 ∘ 𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
11		simpl2 1190	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 ∈ 𝑇)
12	3, 4, 5	ltrn1o 38065	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
13	1, 11, 12	syl2anc 583	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐵–1-1-onto→𝐵)
14		f1of 6700	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
15		fcoi1 6632	. . . . . 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 2779	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = (𝐹 ∘ (◡𝐺 ∘ 𝐺)))
18		coass 6158	. . . 4 ⊢ ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (𝐹 ∘ (◡𝐺 ∘ 𝐺))
19	17, 18	eqtr4di 2797	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = ((𝐹 ∘ ◡𝐺) ∘ 𝐺))
20		simpr 484	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))
21	20	coeq1d 5759	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22		f1of 6700	. . . . 5 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵⟶𝐵)
23		fcoi2 6633	. . . . 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 2778	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = 𝐺)
26	19, 25	eqtrd 2778	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
27		simpr 484	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
28	27	coeq1d 5759	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐺))
29		simpl1 1189	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30		simpl3 1191	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺 ∈ 𝑇)
31	29, 30, 6	syl2anc 583	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺:𝐵–1-1-onto→𝐵)
32		f1ococnv2 6726	. . . 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 2778	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))
35	26, 34	impbida 797	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   I cid 5479  ^◡ccnv 5579   ↾ cres 5582   ∘ ccom 5584  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  Basecbs 16840  HLchlt 37291  LHypclh 37925  LTrncltrn 38042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-laut 37930  df-ldil 38045  df-ltrn 38046

This theorem is referenced by:  tendospcanN  38964