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Theorem nninfalllem1 13422
 Description: Lemma for nninfall 13423. (Contributed by Jim Kingdon, 1-Aug-2022.)
Hypotheses
Ref Expression
nninfall.q (𝜑𝑄 ∈ (2o𝑚))
nninfall.inf (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
nninfall.n (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
nninfalllem1.p (𝜑𝑃 ∈ ℕ)
nninfalllem1.n0 (𝜑 → (𝑄𝑃) = ∅)
Assertion
Ref Expression
nninfalllem1 (𝜑 → ∀𝑛 ∈ ω (𝑃𝑛) = 1o)
Distinct variable groups:   𝑃,𝑖   𝑄,𝑛   𝑖,𝑛,𝜑
Allowed substitution hints:   𝜑(𝑥)   𝑃(𝑥,𝑛)   𝑄(𝑥,𝑖)

Proof of Theorem nninfalllem1
Dummy variables 𝑓 𝑗 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5433 . . . . . 6 (𝑢 = 𝑣 → (𝑃𝑢) = (𝑃𝑣))
21eqeq1d 2150 . . . . 5 (𝑢 = 𝑣 → ((𝑃𝑢) = 1o ↔ (𝑃𝑣) = 1o))
32imbi2d 229 . . . 4 (𝑢 = 𝑣 → ((𝜑 → (𝑃𝑢) = 1o) ↔ (𝜑 → (𝑃𝑣) = 1o)))
4 fveq2 5433 . . . . . 6 (𝑢 = 𝑛 → (𝑃𝑢) = (𝑃𝑛))
54eqeq1d 2150 . . . . 5 (𝑢 = 𝑛 → ((𝑃𝑢) = 1o ↔ (𝑃𝑛) = 1o))
65imbi2d 229 . . . 4 (𝑢 = 𝑛 → ((𝜑 → (𝑃𝑢) = 1o) ↔ (𝜑 → (𝑃𝑛) = 1o)))
7 1n0 6341 . . . . . . . 8 1o ≠ ∅
87nesymi 2356 . . . . . . 7 ¬ ∅ = 1o
9 nninfalllem1.p . . . . . . . . . . . 12 (𝜑𝑃 ∈ ℕ)
109ad2antlr 481 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑃 ∈ ℕ)
11 simplll 523 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑢 ∈ ω)
12 simplr 520 . . . . . . . . . . . 12 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝜑)
13 simpllr 524 . . . . . . . . . . . . 13 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o))
14 r19.21v 2514 . . . . . . . . . . . . 13 (∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o) ↔ (𝜑 → ∀𝑣𝑢 (𝑃𝑣) = 1o))
1513, 14sylib 121 . . . . . . . . . . . 12 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝜑 → ∀𝑣𝑢 (𝑃𝑣) = 1o))
1612, 15mpd 13 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∀𝑣𝑢 (𝑃𝑣) = 1o)
17 simpr 109 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑃𝑢) = ∅)
1810, 11, 16, 17nninfalllemn 13421 . . . . . . . . . 10 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅)))
1918fveq2d 5437 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄𝑃) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅))))
20 nninfalllem1.n0 . . . . . . . . . 10 (𝜑 → (𝑄𝑃) = ∅)
2120ad2antlr 481 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄𝑃) = ∅)
22 elequ2 2117 . . . . . . . . . . . . . 14 (𝑛 = 𝑢 → (𝑖𝑛𝑖𝑢))
2322ifbid 3500 . . . . . . . . . . . . 13 (𝑛 = 𝑢 → if(𝑖𝑛, 1o, ∅) = if(𝑖𝑢, 1o, ∅))
2423mpteq2dv 4029 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅)))
2524fveq2d 5437 . . . . . . . . . . 11 (𝑛 = 𝑢 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅))))
2625eqeq1d 2150 . . . . . . . . . 10 (𝑛 = 𝑢 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅))) = 1o))
27 nninfall.n . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
2827ad2antlr 481 . . . . . . . . . 10 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
2926, 28, 11rspcdva 2800 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅))) = 1o)
3019, 21, 293eqtr3d 2182 . . . . . . . 8 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∅ = 1o)
3130ex 114 . . . . . . 7 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ((𝑃𝑢) = ∅ → ∅ = 1o))
328, 31mtoi 654 . . . . . 6 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ¬ (𝑃𝑢) = ∅)
33 fveq1 5432 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑃 → (𝑓‘suc 𝑗) = (𝑃‘suc 𝑗))
34 fveq1 5432 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑃 → (𝑓𝑗) = (𝑃𝑗))
3533, 34sseq12d 3135 . . . . . . . . . . . . . . 15 (𝑓 = 𝑃 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
3635ralbidv 2440 . . . . . . . . . . . . . 14 (𝑓 = 𝑃 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
37 df-nninf 7022 . . . . . . . . . . . . . 14 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
3836, 37elrab2 2849 . . . . . . . . . . . . 13 (𝑃 ∈ ℕ ↔ (𝑃 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
399, 38sylib 121 . . . . . . . . . . . 12 (𝜑 → (𝑃 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
4039simpld 111 . . . . . . . . . . 11 (𝜑𝑃 ∈ (2o𝑚 ω))
41 elmapi 6576 . . . . . . . . . . 11 (𝑃 ∈ (2o𝑚 ω) → 𝑃:ω⟶2o)
4240, 41syl 14 . . . . . . . . . 10 (𝜑𝑃:ω⟶2o)
4342adantl 275 . . . . . . . . 9 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → 𝑃:ω⟶2o)
44 simpll 519 . . . . . . . . 9 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → 𝑢 ∈ ω)
4543, 44ffvelrnd 5568 . . . . . . . 8 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → (𝑃𝑢) ∈ 2o)
46 elpri 3557 . . . . . . . . 9 ((𝑃𝑢) ∈ {∅, 1o} → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1o))
47 df2o3 6339 . . . . . . . . 9 2o = {∅, 1o}
4846, 47eleq2s 2236 . . . . . . . 8 ((𝑃𝑢) ∈ 2o → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1o))
4945, 48syl 14 . . . . . . 7 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1o))
5049orcomd 719 . . . . . 6 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ((𝑃𝑢) = 1o ∨ (𝑃𝑢) = ∅))
5132, 50ecased 1328 . . . . 5 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → (𝑃𝑢) = 1o)
5251exp31 362 . . . 4 (𝑢 ∈ ω → (∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o) → (𝜑 → (𝑃𝑢) = 1o)))
533, 6, 52omsinds 4547 . . 3 (𝑛 ∈ ω → (𝜑 → (𝑃𝑛) = 1o))
5453impcom 124 . 2 ((𝜑𝑛 ∈ ω) → (𝑃𝑛) = 1o)
5554ralrimiva 2510 1 (𝜑 → ∀𝑛 ∈ ω (𝑃𝑛) = 1o)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1332   ∈ wcel 2112  ∀wral 2418   ⊆ wss 3078  ∅c0 3370  ifcif 3481  {cpr 3535   ↦ cmpt 3999  suc csuc 4298  ωcom 4515  ⟶wf 5131  ‘cfv 5135  (class class class)co 5786  1oc1o 6318  2oc2o 6319   ↑𝑚 cmap 6554  ℕ∞xnninf 7021 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-nul 4064  ax-pow 4108  ax-pr 4142  ax-un 4366  ax-setind 4463  ax-iinf 4513 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-sbc 2916  df-csb 3010  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-nul 3371  df-if 3482  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-int 3782  df-br 3940  df-opab 4000  df-mpt 4001  df-tr 4037  df-id 4226  df-iord 4299  df-on 4301  df-suc 4304  df-iom 4516  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-fv 5143  df-ov 5789  df-oprab 5790  df-mpo 5791  df-1o 6325  df-2o 6326  df-map 6556  df-nninf 7022 This theorem is referenced by:  nninfall  13423
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