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Theorem isopolem 6552
Description: Lemma for isopo 6553. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isopolem (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))

Proof of Theorem isopolem
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 6530 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of 6096 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
3 ffvelrn 6315 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑑𝐴) → (𝐻𝑑) ∈ 𝐵)
43ex 450 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑑𝐴 → (𝐻𝑑) ∈ 𝐵))
5 ffvelrn 6315 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑒𝐴) → (𝐻𝑒) ∈ 𝐵)
65ex 450 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑒𝐴 → (𝐻𝑒) ∈ 𝐵))
7 ffvelrn 6315 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑓𝐴) → (𝐻𝑓) ∈ 𝐵)
87ex 450 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑓𝐴 → (𝐻𝑓) ∈ 𝐵))
94, 6, 83anim123d 1403 . . . . . . . . . . 11 (𝐻:𝐴𝐵 → ((𝑑𝐴𝑒𝐴𝑓𝐴) → ((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵)))
101, 2, 93syl 18 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑑𝐴𝑒𝐴𝑓𝐴) → ((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵)))
1110imp 445 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵))
12 breq12 4620 . . . . . . . . . . . . 13 ((𝑎 = (𝐻𝑑) ∧ 𝑎 = (𝐻𝑑)) → (𝑎𝑆𝑎 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
1312anidms 676 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑎 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
1413notbid 308 . . . . . . . . . . 11 (𝑎 = (𝐻𝑑) → (¬ 𝑎𝑆𝑎 ↔ ¬ (𝐻𝑑)𝑆(𝐻𝑑)))
15 breq1 4618 . . . . . . . . . . . . 13 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑏 ↔ (𝐻𝑑)𝑆𝑏))
1615anbi1d 740 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → ((𝑎𝑆𝑏𝑏𝑆𝑐) ↔ ((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐)))
17 breq1 4618 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑐 ↔ (𝐻𝑑)𝑆𝑐))
1816, 17imbi12d 334 . . . . . . . . . . 11 (𝑎 = (𝐻𝑑) → (((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐) ↔ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐)))
1914, 18anbi12d 746 . . . . . . . . . 10 (𝑎 = (𝐻𝑑) → ((¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐))))
20 breq2 4619 . . . . . . . . . . . . 13 (𝑏 = (𝐻𝑒) → ((𝐻𝑑)𝑆𝑏 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
21 breq1 4618 . . . . . . . . . . . . 13 (𝑏 = (𝐻𝑒) → (𝑏𝑆𝑐 ↔ (𝐻𝑒)𝑆𝑐))
2220, 21anbi12d 746 . . . . . . . . . . . 12 (𝑏 = (𝐻𝑒) → (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐)))
2322imbi1d 331 . . . . . . . . . . 11 (𝑏 = (𝐻𝑒) → ((((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐)))
2423anbi2d 739 . . . . . . . . . 10 (𝑏 = (𝐻𝑒) → ((¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐))))
25 breq2 4619 . . . . . . . . . . . . 13 (𝑐 = (𝐻𝑓) → ((𝐻𝑒)𝑆𝑐 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
2625anbi2d 739 . . . . . . . . . . . 12 (𝑐 = (𝐻𝑓) → (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓))))
27 breq2 4619 . . . . . . . . . . . 12 (𝑐 = (𝐻𝑓) → ((𝐻𝑑)𝑆𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
2826, 27imbi12d 334 . . . . . . . . . . 11 (𝑐 = (𝐻𝑓) → ((((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓))))
2928anbi2d 739 . . . . . . . . . 10 (𝑐 = (𝐻𝑓) → ((¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
3019, 24, 29rspc3v 3310 . . . . . . . . 9 (((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
3111, 30syl 17 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
32 simpl 473 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
33 simpr1 1065 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑑𝐴)
34 isorel 6533 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑑𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
3532, 33, 33, 34syl12anc 1321 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
3635notbid 308 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (¬ 𝑑𝑅𝑑 ↔ ¬ (𝐻𝑑)𝑆(𝐻𝑑)))
37 simpr2 1066 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑒𝐴)
38 isorel 6533 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
3932, 33, 37, 38syl12anc 1321 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
40 simpr3 1067 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑓𝐴)
41 isorel 6533 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑒𝐴𝑓𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
4232, 37, 40, 41syl12anc 1321 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
4339, 42anbi12d 746 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((𝑑𝑅𝑒𝑒𝑅𝑓) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓))))
44 isorel 6533 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑓𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
4532, 33, 40, 44syl12anc 1321 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
4643, 45imbi12d 334 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓))))
4736, 46anbi12d 746 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
4831, 47sylibrd 249 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
4948ex 450 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑑𝐴𝑒𝐴𝑓𝐴) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))))
5049com23 86 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → ((𝑑𝐴𝑒𝐴𝑓𝐴) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))))
5150imp31 448 . . . 4 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐))) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5251ralrimivvva 2966 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐))) → ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5352ex 450 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
54 df-po 4997 . 2 (𝑆 Po 𝐵 ↔ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)))
55 df-po 4997 . 2 (𝑅 Po 𝐴 ↔ ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5653, 54, 553imtr4g 285 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4615   Po wpo 4995  wf 5845  1-1-ontowf1o 5848  cfv 5849   Isom wiso 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-id 4991  df-po 4997  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-f1o 5856  df-fv 5857  df-isom 5858
This theorem is referenced by:  isopo  6553  isosolem  6554
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