Theorem pjss2coi

Description: Subset relationship for projections. Theorem 4.5(i)<->(ii) of Beran p. 112. (Contributed by NM, 7-Oct-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
	pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjss2coi	$$⊢G\subseteq H\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	pjcoi	$$⊢x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)$$
4	3	adantl	$$⊢\left(G\subseteq H\wedge x\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)$$
5		2fveq3	$$⊢x=if\left(x\in \mathscr{H},x,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)\right)$$
6		fveq2	$$⊢x=if\left(x\in \mathscr{H},x,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)$$
7	5 6	eqeq12d	$$⊢x=if\left(x\in \mathscr{H},x,0_{ℎ}\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)\right)$$
8	7	imbi2d	$$⊢x=if\left(x\in \mathscr{H},x,0_{ℎ}\right)\to \left(\left(G\subseteq H\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\right)\leftrightarrow \left(G\subseteq H\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)\right)\right)$$
9		ifhvhv0	$$⊢if\left(x\in \mathscr{H},x,0_{ℎ}\right)\in \mathscr{H}$$
10	1 9 2	pjss2i	$$⊢G\subseteq H\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(x\in \mathscr{H},x,0_{ℎ}\right)\right)$$
11	8 10	dedth	$$⊢x\in \mathscr{H}\to \left(G\subseteq H\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\right)$$
12	11	impcom	$$⊢\left(G\subseteq H\wedge x\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)$$
13	4 12	eqtrd	$$⊢\left(G\subseteq H\wedge x\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)$$
14	13	ralrimiva	$$⊢G\subseteq H\to \forall x\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)$$
15	1	pjfi	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right):\mathscr{H}⟶\mathscr{H}$$
16	2	pjfi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right):\mathscr{H}⟶\mathscr{H}$$
17	15 16	hocofi	$$⊢\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right):\mathscr{H}⟶\mathscr{H}$$
18	17 15	hoeqi	$$⊢\forall x\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)$$
19	14 18	sylib	$$⊢G\subseteq H\to {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)$$
20		fveq1	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(y\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(y\right)$$
21	20	oveq2d	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to x{\cdot }_{\mathrm{ih}}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(y\right)=x{\cdot }_{\mathrm{ih}}{\mathrm{proj}}_{ℎ}\left(G\right)\left(y\right)$$
22	21	ad2antlr	$$⊢\left(\left(x\in \mathscr{H}\wedge {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\right)\wedge y\in \mathscr{H}\right)\to x{\cdot }_{\mathrm{ih}}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(y\right)=x{\cdot }_{\mathrm{ih}}{\mathrm{proj}}_{ℎ}\left(G\right)\left(y\right)$$
23	2 1	pjadjcoi	$$⊢\left(x\in \mathscr{H}\wedge y\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right){\cdot }_{\mathrm{ih}}y=x{\cdot }_{\mathrm{ih}}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(y\right)$$
24	23	adantlr	$$⊢\left(\left(x\in \mathscr{H}\wedge {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\right)\wedge y\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right){\cdot }_{\mathrm{ih}}y=x{\cdot }_{\mathrm{ih}}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(y\right)$$
25	1	pjadji	$$⊢\left(x\in \mathscr{H}\wedge y\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right){\cdot }_{\mathrm{ih}}y=x{\cdot }_{\mathrm{ih}}{\mathrm{proj}}_{ℎ}\left(G\right)\left(y\right)$$
26	25	adantlr	$$⊢\left(\left(x\in \mathscr{H}\wedge {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\right)\wedge y\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right){\cdot }_{\mathrm{ih}}y=x{\cdot }_{\mathrm{ih}}{\mathrm{proj}}_{ℎ}\left(G\right)\left(y\right)$$
27	22 24 26	3eqtr4d	$$⊢\left(\left(x\in \mathscr{H}\wedge {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\right)\wedge y\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right){\cdot }_{\mathrm{ih}}y={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right){\cdot }_{\mathrm{ih}}y$$
28	27	exp31	$$⊢x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to \left(y\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right){\cdot }_{\mathrm{ih}}y={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right){\cdot }_{\mathrm{ih}}y\right)\right)$$
29	28	ralrimdv	$$⊢x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to \forall y\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right){\cdot }_{\mathrm{ih}}y={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right){\cdot }_{\mathrm{ih}}y\right)$$
30	2 1	pjcohcli	$$⊢x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)\in \mathscr{H}$$
31	1	pjhcli	$$⊢x\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\in \mathscr{H}$$
32		hial2eq	$$⊢\left(\left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)\in \mathscr{H}\wedge {\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\in \mathscr{H}\right)\to \left(\forall y\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right){\cdot }_{\mathrm{ih}}y={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right){\cdot }_{\mathrm{ih}}y\leftrightarrow \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\right)$$
33	30 31 32	syl2anc	$$⊢x\in \mathscr{H}\to \left(\forall y\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right){\cdot }_{\mathrm{ih}}y={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right){\cdot }_{\mathrm{ih}}y\leftrightarrow \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\right)$$
34	29 33	sylibd	$$⊢x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\right)$$
35	34	com12	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to \left(x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\right)$$
36	35	ralrimiv	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to \forall x\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)$$
37	16 15	hocofi	$$⊢\left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right):\mathscr{H}⟶\mathscr{H}$$
38	37 15	hoeqi	$$⊢\forall x\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(x\right)\leftrightarrow {\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)={\mathrm{proj}}_{ℎ}\left(G\right)$$
39	36 38	sylib	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)={\mathrm{proj}}_{ℎ}\left(G\right)$$
40	1 2	pjss1coi	$$⊢G\subseteq H\leftrightarrow {\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(G\right)={\mathrm{proj}}_{ℎ}\left(G\right)$$
41	39 40	sylibr	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)\to G\subseteq H$$
42	19 41	impbii	$$⊢G\subseteq H\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(G\right)$$