cdlemg17h

	Ref	Expression
Hypotheses	cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg12.a	$⊢ A = Atoms (K)$
	cdlemg12.h	$⊢ H = LHyp (K)$
	cdlemg12.t	$⊢ T = LTrn (K) (W)$
	cdlemg12b.r	$⊢ R = trL (K) (W)$
Assertion	cdlemg17h	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (S = F (P) \lor S = F (Q))$

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg12.a	$⊢ A = Atoms (K)$
5		cdlemg12.h	$⊢ H = LHyp (K)$
6		cdlemg12.t	$⊢ T = LTrn (K) (W)$
7		cdlemg12b.r	$⊢ R = trL (K) (W)$
8		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to K \in HL$
9		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q))$
10		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
11		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F \in T$
12		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to S \in A$
13	1 4 5 6	ltrncnvat	$⊢ ((K \in HL \land W \in H) \land F \in T \land S \in A) \to F^{-1} (S) \in A$
14	10 11 12 13	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F^{-1} (S) \in A$
15		eqid	$⊢ {Base}_{K} = {Base}_{K}$
16	15 4	atbase	$⊢ F^{-1} (S) \in A \to F^{-1} (S) \in {Base}_{K}$
17	14 16	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F^{-1} (S) \in {Base}_{K}$
18		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \in A$
19		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to Q \in A$
20	15 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
21	8 18 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
22	15 1 5 6	ltrnle	$⊢ ((K \in HL \land W \in H) \land F \in T \land (F^{-1} (S) \in {Base}_{K} \land P \underset{˙}{\lor} Q \in {Base}_{K})) \to (F^{-1} (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow F (F^{-1} (S)) \underset{˙}{\leq} F (P \underset{˙}{\lor} Q))$
23	10 11 17 21 22	syl112anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F^{-1} (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow F (F^{-1} (S)) \underset{˙}{\leq} F (P \underset{˙}{\lor} Q))$
24	15 5 6	ltrn1o	$⊢ ((K \in HL \land W \in H) \land F \in T) \to F : {Base}_{K} \underset{1-1 onto}{⟶} {Base}_{K}$
25	10 11 24	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F : {Base}_{K} \underset{1-1 onto}{⟶} {Base}_{K}$
26	15 4	atbase	$⊢ S \in A \to S \in {Base}_{K}$
27	12 26	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to S \in {Base}_{K}$
28		f1ocnvfv2	$⊢ (F : {Base}_{K} \underset{1-1 onto}{⟶} {Base}_{K} \land S \in {Base}_{K}) \to F (F^{-1} (S)) = S$
29	25 27 28	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (F^{-1} (S)) = S$
30	15 4	atbase	$⊢ P \in A \to P \in {Base}_{K}$
31	18 30	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \in {Base}_{K}$
32	15 4	atbase	$⊢ Q \in A \to Q \in {Base}_{K}$
33	19 32	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to Q \in {Base}_{K}$
34	15 2 5 6	ltrnj	$⊢ ((K \in HL \land W \in H) \land F \in T \land (P \in {Base}_{K} \land Q \in {Base}_{K})) \to F (P \underset{˙}{\lor} Q) = F (P) \underset{˙}{\lor} F (Q)$
35	10 11 31 33 34	syl112anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P \underset{˙}{\lor} Q) = F (P) \underset{˙}{\lor} F (Q)$
36	29 35	breq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F (F^{-1} (S)) \underset{˙}{\leq} F (P \underset{˙}{\lor} Q) \leftrightarrow S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))$
37	23 36	bitr2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)) \leftrightarrow F^{-1} (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
38	9 37	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F^{-1} (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
39		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
40		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \neq Q$
41		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
42	1 4 5 6	ltrncnvel	$⊢ ((K \in HL \land W \in H) \land F \in T \land (S \in A \land \neg S \underset{˙}{\leq} W)) \to (F^{-1} (S) \in A \land \neg F^{-1} (S) \underset{˙}{\leq} W)$
43	10 11 41 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F^{-1} (S) \in A \land \neg F^{-1} (S) \underset{˙}{\leq} W)$
44	1 2 4	cdleme0nex	$⊢ ((K \in HL \land F^{-1} (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)) \land (P \in A \land Q \in A \land P \neq Q) \land (F^{-1} (S) \in A \land \neg F^{-1} (S) \underset{˙}{\leq} W)) \to (F^{-1} (S) = P \lor F^{-1} (S) = Q)$
45	8 38 39 18 19 40 43 44	syl331anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F^{-1} (S) = P \lor F^{-1} (S) = Q)$
46		f1ocnvfvb	$⊢ (F : {Base}_{K} \underset{1-1 onto}{⟶} {Base}_{K} \land P \in {Base}_{K} \land S \in {Base}_{K}) \to (F (P) = S \leftrightarrow F^{-1} (S) = P)$
47	25 31 27 46	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F (P) = S \leftrightarrow F^{-1} (S) = P)$
48		eqcom	$⊢ F (P) = S \leftrightarrow S = F (P)$
49	47 48	bitr3di	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F^{-1} (S) = P \leftrightarrow S = F (P))$
50		f1ocnvfvb	$⊢ (F : {Base}_{K} \underset{1-1 onto}{⟶} {Base}_{K} \land Q \in {Base}_{K} \land S \in {Base}_{K}) \to (F (Q) = S \leftrightarrow F^{-1} (S) = Q)$
51	25 33 27 50	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F (Q) = S \leftrightarrow F^{-1} (S) = Q)$
52		eqcom	$⊢ F (Q) = S \leftrightarrow S = F (Q)$
53	51 52	bitr3di	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F^{-1} (S) = Q \leftrightarrow S = F (Q))$
54	49 53	orbi12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to ((F^{-1} (S) = P \lor F^{-1} (S) = Q) \leftrightarrow (S = F (P) \lor S = F (Q)))$
55	45 54	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land (P \neq Q \land S \underset{˙}{\leq} (F (P) \underset{˙}{\lor} F (Q)))) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (S = F (P) \lor S = F (Q))$

Metamath Proof Explorer

Theorem cdlemg17h

Proof