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Theorem nninfalllem1 14760
Description: Lemma for nninfall 14761. (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 5516 . . . . . 6 (𝑢 = 𝑣 → (𝑃𝑢) = (𝑃𝑣))
21eqeq1d 2186 . . . . 5 (𝑢 = 𝑣 → ((𝑃𝑢) = 1o ↔ (𝑃𝑣) = 1o))
32imbi2d 230 . . . 4 (𝑢 = 𝑣 → ((𝜑 → (𝑃𝑢) = 1o) ↔ (𝜑 → (𝑃𝑣) = 1o)))
4 fveq2 5516 . . . . . 6 (𝑢 = 𝑛 → (𝑃𝑢) = (𝑃𝑛))
54eqeq1d 2186 . . . . 5 (𝑢 = 𝑛 → ((𝑃𝑢) = 1o ↔ (𝑃𝑛) = 1o))
65imbi2d 230 . . . 4 (𝑢 = 𝑛 → ((𝜑 → (𝑃𝑢) = 1o) ↔ (𝜑 → (𝑃𝑛) = 1o)))
7 1n0 6433 . . . . . . . 8 1o ≠ ∅
87nesymi 2393 . . . . . . 7 ¬ ∅ = 1o
9 nninfalllem1.p . . . . . . . . . . . 12 (𝜑𝑃 ∈ ℕ)
109ad2antlr 489 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑃 ∈ ℕ)
11 simplll 533 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑢 ∈ ω)
12 simplr 528 . . . . . . . . . . . 12 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝜑)
13 simpllr 534 . . . . . . . . . . . . 13 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o))
14 r19.21v 2554 . . . . . . . . . . . . 13 (∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o) ↔ (𝜑 → ∀𝑣𝑢 (𝑃𝑣) = 1o))
1513, 14sylib 122 . . . . . . . . . . . 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 110 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑃𝑢) = ∅)
1810, 11, 16, 17nnnninfeq 7126 . . . . . . . . . 10 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅)))
1918fveq2d 5520 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄𝑃) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅))))
20 nninfalllem1.n0 . . . . . . . . . 10 (𝜑 → (𝑄𝑃) = ∅)
2120ad2antlr 489 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄𝑃) = ∅)
22 elequ2 2153 . . . . . . . . . . . . . 14 (𝑛 = 𝑢 → (𝑖𝑛𝑖𝑢))
2322ifbid 3556 . . . . . . . . . . . . 13 (𝑛 = 𝑢 → if(𝑖𝑛, 1o, ∅) = if(𝑖𝑢, 1o, ∅))
2423mpteq2dv 4095 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅)))
2524fveq2d 5520 . . . . . . . . . . 11 (𝑛 = 𝑢 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅))))
2625eqeq1d 2186 . . . . . . . . . 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 489 . . . . . . . . . 10 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
2926, 28, 11rspcdva 2847 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1o, ∅))) = 1o)
3019, 21, 293eqtr3d 2218 . . . . . . . 8 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∅ = 1o)
3130ex 115 . . . . . . 7 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ((𝑃𝑢) = ∅ → ∅ = 1o))
328, 31mtoi 664 . . . . . 6 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ¬ (𝑃𝑢) = ∅)
33 fveq1 5515 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑃 → (𝑓‘suc 𝑗) = (𝑃‘suc 𝑗))
34 fveq1 5515 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑃 → (𝑓𝑗) = (𝑃𝑗))
3533, 34sseq12d 3187 . . . . . . . . . . . . . . 15 (𝑓 = 𝑃 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
3635ralbidv 2477 . . . . . . . . . . . . . 14 (𝑓 = 𝑃 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
37 df-nninf 7119 . . . . . . . . . . . . . 14 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
3836, 37elrab2 2897 . . . . . . . . . . . . 13 (𝑃 ∈ ℕ ↔ (𝑃 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
399, 38sylib 122 . . . . . . . . . . . 12 (𝜑 → (𝑃 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
4039simpld 112 . . . . . . . . . . 11 (𝜑𝑃 ∈ (2o𝑚 ω))
41 elmapi 6670 . . . . . . . . . . 11 (𝑃 ∈ (2o𝑚 ω) → 𝑃:ω⟶2o)
4240, 41syl 14 . . . . . . . . . 10 (𝜑𝑃:ω⟶2o)
4342adantl 277 . . . . . . . . 9 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → 𝑃:ω⟶2o)
44 simpll 527 . . . . . . . . 9 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → 𝑢 ∈ ω)
4543, 44ffvelcdmd 5653 . . . . . . . 8 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → (𝑃𝑢) ∈ 2o)
46 elpri 3616 . . . . . . . . 9 ((𝑃𝑢) ∈ {∅, 1o} → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1o))
47 df2o3 6431 . . . . . . . . 9 2o = {∅, 1o}
4846, 47eleq2s 2272 . . . . . . . 8 ((𝑃𝑢) ∈ 2o → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1o))
4945, 48syl 14 . . . . . . 7 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1o))
5049orcomd 729 . . . . . 6 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → ((𝑃𝑢) = 1o ∨ (𝑃𝑢) = ∅))
5132, 50ecased 1349 . . . . 5 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o)) ∧ 𝜑) → (𝑃𝑢) = 1o)
5251exp31 364 . . . 4 (𝑢 ∈ ω → (∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1o) → (𝜑 → (𝑃𝑢) = 1o)))
533, 6, 52omsinds 4622 . . 3 (𝑛 ∈ ω → (𝜑 → (𝑃𝑛) = 1o))
5453impcom 125 . 2 ((𝜑𝑛 ∈ ω) → (𝑃𝑛) = 1o)
5554ralrimiva 2550 1 (𝜑 → ∀𝑛 ∈ ω (𝑃𝑛) = 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708   = wceq 1353  wcel 2148  wral 2455  wss 3130  c0 3423  ifcif 3535  {cpr 3594  cmpt 4065  suc csuc 4366  ωcom 4590  wf 5213  cfv 5217  (class class class)co 5875  1oc1o 6410  2oc2o 6411  𝑚 cmap 6648  xnninf 7118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1o 6417  df-2o 6418  df-map 6650  df-nninf 7119
This theorem is referenced by:  nninfall  14761
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