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Theorem isopolem 7090
Description: Lemma for isopo 7091. (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 7068 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of 6608 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
3 ffvelrn 6842 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑑𝐴) → (𝐻𝑑) ∈ 𝐵)
43ex 415 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑑𝐴 → (𝐻𝑑) ∈ 𝐵))
5 ffvelrn 6842 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑒𝐴) → (𝐻𝑒) ∈ 𝐵)
65ex 415 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑒𝐴 → (𝐻𝑒) ∈ 𝐵))
7 ffvelrn 6842 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑓𝐴) → (𝐻𝑓) ∈ 𝐵)
87ex 415 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑓𝐴 → (𝐻𝑓) ∈ 𝐵))
94, 6, 83anim123d 1437 . . . . . . . . . . 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 409 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵))
12 breq12 5062 . . . . . . . . . . . . 13 ((𝑎 = (𝐻𝑑) ∧ 𝑎 = (𝐻𝑑)) → (𝑎𝑆𝑎 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
1312anidms 569 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑎 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
1413notbid 320 . . . . . . . . . . 11 (𝑎 = (𝐻𝑑) → (¬ 𝑎𝑆𝑎 ↔ ¬ (𝐻𝑑)𝑆(𝐻𝑑)))
15 breq1 5060 . . . . . . . . . . . . 13 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑏 ↔ (𝐻𝑑)𝑆𝑏))
1615anbi1d 631 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → ((𝑎𝑆𝑏𝑏𝑆𝑐) ↔ ((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐)))
17 breq1 5060 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑐 ↔ (𝐻𝑑)𝑆𝑐))
1816, 17imbi12d 347 . . . . . . . . . . 11 (𝑎 = (𝐻𝑑) → (((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐) ↔ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐)))
1914, 18anbi12d 632 . . . . . . . . . 10 (𝑎 = (𝐻𝑑) → ((¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐))))
20 breq2 5061 . . . . . . . . . . . . 13 (𝑏 = (𝐻𝑒) → ((𝐻𝑑)𝑆𝑏 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
21 breq1 5060 . . . . . . . . . . . . 13 (𝑏 = (𝐻𝑒) → (𝑏𝑆𝑐 ↔ (𝐻𝑒)𝑆𝑐))
2220, 21anbi12d 632 . . . . . . . . . . . 12 (𝑏 = (𝐻𝑒) → (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐)))
2322imbi1d 344 . . . . . . . . . . 11 (𝑏 = (𝐻𝑒) → ((((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐)))
2423anbi2d 630 . . . . . . . . . 10 (𝑏 = (𝐻𝑒) → ((¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐))))
25 breq2 5061 . . . . . . . . . . . . 13 (𝑐 = (𝐻𝑓) → ((𝐻𝑒)𝑆𝑐 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
2625anbi2d 630 . . . . . . . . . . . 12 (𝑐 = (𝐻𝑓) → (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓))))
27 breq2 5061 . . . . . . . . . . . 12 (𝑐 = (𝐻𝑓) → ((𝐻𝑑)𝑆𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
2826, 27imbi12d 347 . . . . . . . . . . 11 (𝑐 = (𝐻𝑓) → ((((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓))))
2928anbi2d 630 . . . . . . . . . 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 3634 . . . . . . . . 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 485 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
33 simpr1 1189 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑑𝐴)
34 isorel 7071 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑑𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
3532, 33, 33, 34syl12anc 834 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
3635notbid 320 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (¬ 𝑑𝑅𝑑 ↔ ¬ (𝐻𝑑)𝑆(𝐻𝑑)))
37 simpr2 1190 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑒𝐴)
38 isorel 7071 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
3932, 33, 37, 38syl12anc 834 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
40 simpr3 1191 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑓𝐴)
41 isorel 7071 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑒𝐴𝑓𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
4232, 37, 40, 41syl12anc 834 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
4339, 42anbi12d 632 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((𝑑𝑅𝑒𝑒𝑅𝑓) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓))))
44 isorel 7071 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑓𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
4532, 33, 40, 44syl12anc 834 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
4643, 45imbi12d 347 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓))))
4736, 46anbi12d 632 . . . . . . . 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 261 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
4948ex 415 . . . . . 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 420 . . . 4 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐))) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5251ralrimivvva 3190 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐))) → ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5352ex 415 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
54 df-po 5467 . 2 (𝑆 Po 𝐵 ↔ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)))
55 df-po 5467 . 2 (𝑅 Po 𝐴 ↔ ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5653, 54, 553imtr4g 298 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 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136   class class class wbr 5057   Po wpo 5465  wf 6344  1-1-ontowf1o 6347  cfv 6348   Isom wiso 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-po 5467  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-f1o 6355  df-fv 6356  df-isom 6357
This theorem is referenced by:  isopo  7091  isosolem  7092
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