Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj554 Structured version   Visualization version   GIF version

Theorem bnj554 34892
Description: Technical lemma for bnj852 34914. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj554.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj554.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj554.21 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.22 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj554.23 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.24 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
Assertion
Ref Expression
bnj554 ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
Distinct variable groups:   𝑦,𝐺   𝑦,𝑖   𝑦,𝑝
Allowed substitution hints:   𝜂(𝑦,𝑖,𝑚,𝑛,𝑝)   𝜁(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐺(𝑖,𝑚,𝑛,𝑝)   𝐾(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐿(𝑦,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj554
StepHypRef Expression
1 bnj554.19 . . 3 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
21bnj1254 34802 . 2 (𝜂𝑚 = suc 𝑝)
3 bnj554.20 . . 3 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
43simp3bi 1146 . 2 (𝜁𝑚 = suc 𝑖)
5 simpr 484 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑚 = suc 𝑖)
6 bnj551 34735 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
7 fveq2 6907 . . . 4 (𝑚 = suc 𝑖 → (𝐺𝑚) = (𝐺‘suc 𝑖))
8 fveq2 6907 . . . . 5 (𝑝 = 𝑖 → (𝐺𝑝) = (𝐺𝑖))
9 iuneq1 5013 . . . . . 6 ((𝐺𝑝) = (𝐺𝑖) → 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
10 bnj554.24 . . . . . 6 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
11 bnj554.23 . . . . . 6 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
129, 10, 113eqtr4g 2800 . . . . 5 ((𝐺𝑝) = (𝐺𝑖) → 𝐿 = 𝐾)
138, 12syl 17 . . . 4 (𝑝 = 𝑖𝐿 = 𝐾)
147, 13eqeqan12d 2749 . . 3 ((𝑚 = suc 𝑖𝑝 = 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
155, 6, 14syl2anc 584 . 2 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
162, 4, 15syl2an 596 1 ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106   ciun 4996  suc csuc 6388  cfv 6563  ωcom 7887  w-bnj17 34679   predc-bnj14 34681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641  df-suc 6392  df-iota 6516  df-fv 6571  df-bnj17 34680
This theorem is referenced by:  bnj558  34895
  Copyright terms: Public domain W3C validator