MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgr3cyclexlem Structured version   Visualization version   GIF version

Theorem uhgr3cyclexlem 27525
Description: Lemma for uhgr3cyclex 27526. (Contributed by AV, 12-Feb-2021.)
Hypotheses
Ref Expression
uhgr3cyclex.v 𝑉 = (Vtx‘𝐺)
uhgr3cyclex.e 𝐸 = (Edg‘𝐺)
uhgr3cyclex.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
uhgr3cyclexlem ((((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → 𝐽𝐾)

Proof of Theorem uhgr3cyclexlem
StepHypRef Expression
1 fveq2 6411 . . . . . . . . 9 (𝐽 = 𝐾 → (𝐼𝐽) = (𝐼𝐾))
21eqeq2d 2809 . . . . . . . 8 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) ↔ {𝐵, 𝐶} = (𝐼𝐾)))
3 eqeq2 2810 . . . . . . . . . . . 12 ((𝐼𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
43eqcoms 2807 . . . . . . . . . . 11 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5 prcom 4456 . . . . . . . . . . . . . 14 {𝐶, 𝐴} = {𝐴, 𝐶}
65eqeq1i 2804 . . . . . . . . . . . . 13 ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7 simpl 475 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
8 simpr 478 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
97, 8preq1b 4563 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
109biimpcd 241 . . . . . . . . . . . . 13 ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
116, 10sylbi 209 . . . . . . . . . . . 12 ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
1211eqcoms 2807 . . . . . . . . . . 11 ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
134, 12syl6bi 245 . . . . . . . . . 10 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
1413adantl 474 . . . . . . . . 9 ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
1514com12 32 . . . . . . . 8 ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
162, 15syl6bi 245 . . . . . . 7 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))))
1716adantld 485 . . . . . 6 (𝐽 = 𝐾 → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))))
1817com14 96 . . . . 5 ((𝐴𝑉𝐵𝑉) → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → (𝐽 = 𝐾𝐴 = 𝐵))))
1918imp32 410 . . . 4 (((𝐴𝑉𝐵𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → (𝐽 = 𝐾𝐴 = 𝐵))
2019necon3d 2992 . . 3 (((𝐴𝑉𝐵𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → (𝐴𝐵𝐽𝐾))
2120impancom 444 . 2 (((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾))) → 𝐽𝐾))
2221imp 396 1 ((((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → 𝐽𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wne 2971  {cpr 4370  dom cdm 5312  cfv 6101  Vtxcvtx 26231  iEdgciedg 26232  Edgcedg 26282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109
This theorem is referenced by:  uhgr3cyclex  27526
  Copyright terms: Public domain W3C validator