Metamath Proof Explorer


Theorem cdlemg2cex

Description: Any translation is one of our F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf ? (Contributed by NM, 22-Apr-2013)

Ref Expression
Hypotheses cdlemg2.b 𝐵 = ( Base ‘ 𝐾 )
cdlemg2.l = ( le ‘ 𝐾 )
cdlemg2.j = ( join ‘ 𝐾 )
cdlemg2.m = ( meet ‘ 𝐾 )
cdlemg2.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg2.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg2.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg2ex.u 𝑈 = ( ( 𝑝 𝑞 ) 𝑊 )
cdlemg2ex.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑞 ( ( 𝑝 𝑡 ) 𝑊 ) ) )
cdlemg2ex.e 𝐸 = ( ( 𝑝 𝑞 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdlemg2ex.g 𝐺 = ( 𝑥𝐵 ↦ if ( ( 𝑝𝑞 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑝 𝑞 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑝 𝑞 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
Assertion cdlemg2cex ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝐹𝑇 ↔ ∃ 𝑝𝐴𝑞𝐴 ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg2.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemg2.l = ( le ‘ 𝐾 )
3 cdlemg2.j = ( join ‘ 𝐾 )
4 cdlemg2.m = ( meet ‘ 𝐾 )
5 cdlemg2.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemg2.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemg2.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemg2ex.u 𝑈 = ( ( 𝑝 𝑞 ) 𝑊 )
9 cdlemg2ex.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑞 ( ( 𝑝 𝑡 ) 𝑊 ) ) )
10 cdlemg2ex.e 𝐸 = ( ( 𝑝 𝑞 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
11 cdlemg2ex.g 𝐺 = ( 𝑥𝐵 ↦ if ( ( 𝑝𝑞 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑝 𝑞 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑝 𝑞 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
12 2 5 6 7 cdlemg1cex ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝐹𝑇 ↔ ∃ 𝑝𝐴𝑞𝐴 ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) ) ) )
13 simplll ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → 𝐾 ∈ HL )
14 simpllr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → 𝑊𝐻 )
15 simplrl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → 𝑝𝐴 )
16 simprl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → ¬ 𝑝 𝑊 )
17 simplrr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → 𝑞𝐴 )
18 simprr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → ¬ 𝑞 𝑊 )
19 eqid ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 )
20 1 2 3 4 5 6 8 9 10 11 7 19 cdlemg1b2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ∧ ( 𝑞𝐴 ∧ ¬ 𝑞 𝑊 ) ) → ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) = 𝐺 )
21 13 14 15 16 17 18 20 syl222anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) = 𝐺 )
22 21 eqeq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) ∧ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) → ( 𝐹 = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) ↔ 𝐹 = 𝐺 ) )
23 22 pm5.32da ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) → ( ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ∧ 𝐹 = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) ) ↔ ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ∧ 𝐹 = 𝐺 ) ) )
24 df-3an ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) ) ↔ ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ∧ 𝐹 = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) ) )
25 df-3an ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺 ) ↔ ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ∧ 𝐹 = 𝐺 ) )
26 23 24 25 3bitr4g ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑝𝐴𝑞𝐴 ) ) → ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) ) ↔ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺 ) ) )
27 26 2rexbidva ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( ∃ 𝑝𝐴𝑞𝐴 ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = ( 𝑓𝑇 ( 𝑓𝑝 ) = 𝑞 ) ) ↔ ∃ 𝑝𝐴𝑞𝐴 ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺 ) ) )
28 12 27 bitrd ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝐹𝑇 ↔ ∃ 𝑝𝐴𝑞𝐴 ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺 ) ) )