# Metamath Proof Explorer

## Theorem mdbr

Description: Binary relation expressing <. A , B >. is a modular pair. Definition 1.1 of MaedaMaeda p. 1. (Contributed by NM, 14-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion mdbr ${⊢}\left({A}\in {\mathbf{C}}_{ℋ}\wedge {B}\in {\mathbf{C}}_{ℋ}\right)\to \left({A}{𝑀}_{ℋ}{B}↔\forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {B}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {B}={x}{\vee }_{ℋ}\left({A}\cap {B}\right)\right)\right)$

### Proof

Step Hyp Ref Expression
1 eleq1 ${⊢}{y}={A}\to \left({y}\in {\mathbf{C}}_{ℋ}↔{A}\in {\mathbf{C}}_{ℋ}\right)$
2 1 anbi1d ${⊢}{y}={A}\to \left(\left({y}\in {\mathbf{C}}_{ℋ}\wedge {z}\in {\mathbf{C}}_{ℋ}\right)↔\left({A}\in {\mathbf{C}}_{ℋ}\wedge {z}\in {\mathbf{C}}_{ℋ}\right)\right)$
3 oveq2 ${⊢}{y}={A}\to {x}{\vee }_{ℋ}{y}={x}{\vee }_{ℋ}{A}$
4 3 ineq1d ${⊢}{y}={A}\to \left({x}{\vee }_{ℋ}{y}\right)\cap {z}=\left({x}{\vee }_{ℋ}{A}\right)\cap {z}$
5 ineq1 ${⊢}{y}={A}\to {y}\cap {z}={A}\cap {z}$
6 5 oveq2d ${⊢}{y}={A}\to {x}{\vee }_{ℋ}\left({y}\cap {z}\right)={x}{\vee }_{ℋ}\left({A}\cap {z}\right)$
7 4 6 eqeq12d ${⊢}{y}={A}\to \left(\left({x}{\vee }_{ℋ}{y}\right)\cap {z}={x}{\vee }_{ℋ}\left({y}\cap {z}\right)↔\left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)\right)$
8 7 imbi2d ${⊢}{y}={A}\to \left(\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{y}\right)\cap {z}={x}{\vee }_{ℋ}\left({y}\cap {z}\right)\right)↔\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)\right)\right)$
9 8 ralbidv ${⊢}{y}={A}\to \left(\forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{y}\right)\cap {z}={x}{\vee }_{ℋ}\left({y}\cap {z}\right)\right)↔\forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)\right)\right)$
10 2 9 anbi12d ${⊢}{y}={A}\to \left(\left(\left({y}\in {\mathbf{C}}_{ℋ}\wedge {z}\in {\mathbf{C}}_{ℋ}\right)\wedge \forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{y}\right)\cap {z}={x}{\vee }_{ℋ}\left({y}\cap {z}\right)\right)\right)↔\left(\left({A}\in {\mathbf{C}}_{ℋ}\wedge {z}\in {\mathbf{C}}_{ℋ}\right)\wedge \forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)\right)\right)\right)$
11 eleq1 ${⊢}{z}={B}\to \left({z}\in {\mathbf{C}}_{ℋ}↔{B}\in {\mathbf{C}}_{ℋ}\right)$
12 11 anbi2d ${⊢}{z}={B}\to \left(\left({A}\in {\mathbf{C}}_{ℋ}\wedge {z}\in {\mathbf{C}}_{ℋ}\right)↔\left({A}\in {\mathbf{C}}_{ℋ}\wedge {B}\in {\mathbf{C}}_{ℋ}\right)\right)$
13 sseq2 ${⊢}{z}={B}\to \left({x}\subseteq {z}↔{x}\subseteq {B}\right)$
14 ineq2 ${⊢}{z}={B}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {z}=\left({x}{\vee }_{ℋ}{A}\right)\cap {B}$
15 ineq2 ${⊢}{z}={B}\to {A}\cap {z}={A}\cap {B}$
16 15 oveq2d ${⊢}{z}={B}\to {x}{\vee }_{ℋ}\left({A}\cap {z}\right)={x}{\vee }_{ℋ}\left({A}\cap {B}\right)$
17 14 16 eqeq12d ${⊢}{z}={B}\to \left(\left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)↔\left({x}{\vee }_{ℋ}{A}\right)\cap {B}={x}{\vee }_{ℋ}\left({A}\cap {B}\right)\right)$
18 13 17 imbi12d ${⊢}{z}={B}\to \left(\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)\right)↔\left({x}\subseteq {B}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {B}={x}{\vee }_{ℋ}\left({A}\cap {B}\right)\right)\right)$
19 18 ralbidv ${⊢}{z}={B}\to \left(\forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)\right)↔\forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {B}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {B}={x}{\vee }_{ℋ}\left({A}\cap {B}\right)\right)\right)$
20 12 19 anbi12d ${⊢}{z}={B}\to \left(\left(\left({A}\in {\mathbf{C}}_{ℋ}\wedge {z}\in {\mathbf{C}}_{ℋ}\right)\wedge \forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {z}={x}{\vee }_{ℋ}\left({A}\cap {z}\right)\right)\right)↔\left(\left({A}\in {\mathbf{C}}_{ℋ}\wedge {B}\in {\mathbf{C}}_{ℋ}\right)\wedge \forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {B}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {B}={x}{\vee }_{ℋ}\left({A}\cap {B}\right)\right)\right)\right)$
21 df-md ${⊢}{𝑀}_{ℋ}=\left\{⟨{y},{z}⟩|\left(\left({y}\in {\mathbf{C}}_{ℋ}\wedge {z}\in {\mathbf{C}}_{ℋ}\right)\wedge \forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {z}\to \left({x}{\vee }_{ℋ}{y}\right)\cap {z}={x}{\vee }_{ℋ}\left({y}\cap {z}\right)\right)\right)\right\}$
22 10 20 21 brabg ${⊢}\left({A}\in {\mathbf{C}}_{ℋ}\wedge {B}\in {\mathbf{C}}_{ℋ}\right)\to \left({A}{𝑀}_{ℋ}{B}↔\left(\left({A}\in {\mathbf{C}}_{ℋ}\wedge {B}\in {\mathbf{C}}_{ℋ}\right)\wedge \forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {B}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {B}={x}{\vee }_{ℋ}\left({A}\cap {B}\right)\right)\right)\right)$
23 22 bianabs ${⊢}\left({A}\in {\mathbf{C}}_{ℋ}\wedge {B}\in {\mathbf{C}}_{ℋ}\right)\to \left({A}{𝑀}_{ℋ}{B}↔\forall {x}\in {\mathbf{C}}_{ℋ}\phantom{\rule{.4em}{0ex}}\left({x}\subseteq {B}\to \left({x}{\vee }_{ℋ}{A}\right)\cap {B}={x}{\vee }_{ℋ}\left({A}\cap {B}\right)\right)\right)$