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 31297
Description: Technical lemma for bnj852 31319. 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 31208 . 2 (𝜂𝑚 = suc 𝑝)
3 bnj554.20 . . 3 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
43simp3bi 1170 . 2 (𝜁𝑚 = suc 𝑖)
5 simpr 473 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑚 = suc 𝑖)
6 bnj551 31140 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
7 fveq2 6411 . . . 4 (𝑚 = suc 𝑖 → (𝐺𝑚) = (𝐺‘suc 𝑖))
8 fveq2 6411 . . . . 5 (𝑝 = 𝑖 → (𝐺𝑝) = (𝐺𝑖))
9 iuneq1 4733 . . . . . 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 2872 . . . . 5 ((𝐺𝑝) = (𝐺𝑖) → 𝐿 = 𝐾)
138, 12syl 17 . . . 4 (𝑝 = 𝑖𝐿 = 𝐾)
147, 13eqeqan12d 2829 . . 3 ((𝑚 = suc 𝑖𝑝 = 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
155, 6, 14syl2anc 575 . 2 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
162, 4, 15syl2an 585 1 ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157   ciun 4719  suc csuc 5945  cfv 6104  ωcom 7298  w-bnj17 31083   predc-bnj14 31085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103  ax-un 7182  ax-reg 8739
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-eprel 5231  df-fr 5277  df-suc 5949  df-iota 6067  df-fv 6112  df-bnj17 31084
This theorem is referenced by:  bnj558  31300
  Copyright terms: Public domain W3C validator