Theorem ltrncoidN 40111

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 1190	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1192	. . . . . . . 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 40107	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:𝐵–1-1-onto→𝐵)
7	1, 2, 6	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺:𝐵–1-1-onto→𝐵)
8		f1ococnv1 6878	. . . . . . 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 5876	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → (𝐹 ∘ (◡𝐺 ∘ 𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
11		simpl2 1191	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 ∈ 𝑇)
12	3, 4, 5	ltrn1o 40107	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
13	1, 11, 12	syl2anc 584	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐵–1-1-onto→𝐵)
14		f1of 6849	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
15		fcoi1 6783	. . . . . 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 2776	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = (𝐹 ∘ (◡𝐺 ∘ 𝐺)))
18		coass 6287	. . . 4 ⊢ ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (𝐹 ∘ (◡𝐺 ∘ 𝐺))
19	17, 18	eqtr4di 2793	. . 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 5875	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22		f1of 6849	. . . . 5 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵⟶𝐵)
23		fcoi2 6784	. . . . 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 2775	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → ((𝐹 ∘ ◡𝐺) ∘ 𝐺) = 𝐺)
26	19, 25	eqtrd 2775	. 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 5875	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐺))
29		simpl1 1190	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30		simpl3 1192	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺 ∈ 𝑇)
31	29, 30, 6	syl2anc 584	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → 𝐺:𝐵–1-1-onto→𝐵)
32		f1ococnv2 6876	. . . 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 2775	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = 𝐺) → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))
35	26, 34	impbida 801	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) ↔ 𝐹 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   I cid 5582  ^◡ccnv 5688   ↾ cres 5691   ∘ ccom 5693  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  Basecbs 17245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-laut 39972  df-ldil 40087  df-ltrn 40088

This theorem is referenced by:  tendospcanN  41006