cdlemn9

Description: Part of proof of Lemma N of Crawley p. 121 line 36. (Contributed by NM, 27-Feb-2014)

	Ref	Expression
Hypotheses	cdlemn8.b	$⊢ B = {Base}_{K}$
	cdlemn8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemn8.a	$⊢ A = Atoms (K)$
	cdlemn8.h	$⊢ H = LHyp (K)$
	cdlemn8.p	$⊢ P = oc (K) (W)$
	cdlemn8.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
	cdlemn8.t	$⊢ T = LTrn (K) (W)$
	cdlemn8.e	$⊢ E = TEndo (K) (W)$
	cdlemn8.u	$⊢ U = DVecH (K) (W)$
	cdlemn8.s	$⊢ \underset{˙}{+} = +_{U}$
	cdlemn8.f	$⊢ F = (ι h \in T \| h (P) = Q)$
	cdlemn8.g	$⊢ G = (ι h \in T \| h (P) = R)$
Assertion	cdlemn9	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to g (Q) = R$

Proof

Step	Hyp	Ref	Expression
1		cdlemn8.b	$⊢ B = {Base}_{K}$
2		cdlemn8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemn8.a	$⊢ A = Atoms (K)$
4		cdlemn8.h	$⊢ H = LHyp (K)$
5		cdlemn8.p	$⊢ P = oc (K) (W)$
6		cdlemn8.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
7		cdlemn8.t	$⊢ T = LTrn (K) (W)$
8		cdlemn8.e	$⊢ E = TEndo (K) (W)$
9		cdlemn8.u	$⊢ U = DVecH (K) (W)$
10		cdlemn8.s	$⊢ \underset{˙}{+} = +_{U}$
11		cdlemn8.f	$⊢ F = (ι h \in T \| h (P) = Q)$
12		cdlemn8.g	$⊢ G = (ι h \in T \| h (P) = R)$
13	1 2 3 4 5 6 7 8 9 10 11 12	cdlemn8	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to g = G \circ F^{-1}$
14	13	fveq1d	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to g (Q) = (G \circ F^{-1}) (Q)$
15		simp1	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (K \in HL \land W \in H)$
16	2 3 4 5	lhpocnel2	$⊢ (K \in HL \land W \in H) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17	16	3ad2ant1	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simp2l	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
19	2 3 4 7 11	ltrniotacl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to F \in T$
20	15 17 18 19	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to F \in T$
21	1 4 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land F \in T) \to F : B \underset{1-1 onto}{⟶} B$
22	15 20 21	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to F : B \underset{1-1 onto}{⟶} B$
23		f1ocnv	$⊢ F : B \underset{1-1 onto}{⟶} B \to F^{-1} : B \underset{1-1 onto}{⟶} B$
24		f1of	$⊢ F^{-1} : B \underset{1-1 onto}{⟶} B \to F^{-1} : B ⟶ B$
25	22 23 24	3syl	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to F^{-1} : B ⟶ B$
26		simp2ll	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to Q \in A$
27	1 3	atbase	$⊢ Q \in A \to Q \in B$
28	26 27	syl	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to Q \in B$
29		fvco3	$⊢ (F^{-1} : B ⟶ B \land Q \in B) \to (G \circ F^{-1}) (Q) = G (F^{-1} (Q))$
30	25 28 29	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (G \circ F^{-1}) (Q) = G (F^{-1} (Q))$
31	2 3 4 7 11	ltrniotacnvval	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to F^{-1} (Q) = P$
32	15 17 18 31	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to F^{-1} (Q) = P$
33	32	fveq2d	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to G (F^{-1} (Q)) = G (P)$
34		simp2r	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
35	2 3 4 7 12	ltrniotaval	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \to G (P) = R$
36	15 17 34 35	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to G (P) = R$
37	33 36	eqtrd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to G (F^{-1} (Q)) = R$
38	14 30 37	3eqtrd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨G, {I ↾}_{T}⟩ = ⟨s (F), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to g (Q) = R$

Metamath Proof Explorer

Theorem cdlemn9

Proof