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Theorem nninfalllem1 11543
Description: Lemma for nninfall 11544. (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 5289 . . . . . 6 (𝑢 = 𝑣 → (𝑃𝑢) = (𝑃𝑣))
21eqeq1d 2096 . . . . 5 (𝑢 = 𝑣 → ((𝑃𝑢) = 1𝑜 ↔ (𝑃𝑣) = 1𝑜))
32imbi2d 228 . . . 4 (𝑢 = 𝑣 → ((𝜑 → (𝑃𝑢) = 1𝑜) ↔ (𝜑 → (𝑃𝑣) = 1𝑜)))
4 fveq2 5289 . . . . . 6 (𝑢 = 𝑛 → (𝑃𝑢) = (𝑃𝑛))
54eqeq1d 2096 . . . . 5 (𝑢 = 𝑛 → ((𝑃𝑢) = 1𝑜 ↔ (𝑃𝑛) = 1𝑜))
65imbi2d 228 . . . 4 (𝑢 = 𝑛 → ((𝜑 → (𝑃𝑢) = 1𝑜) ↔ (𝜑 → (𝑃𝑛) = 1𝑜)))
7 1n0 6179 . . . . . . . 8 1𝑜 ≠ ∅
87nesymi 2301 . . . . . . 7 ¬ ∅ = 1𝑜
9 nninfalllem1.p . . . . . . . . . . . 12 (𝜑𝑃 ∈ ℕ)
109ad2antlr 473 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑃 ∈ ℕ)
11 simplll 500 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑢 ∈ ω)
12 simplr 497 . . . . . . . . . . . 12 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝜑)
13 simpllr 501 . . . . . . . . . . . . 13 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜))
14 r19.21v 2450 . . . . . . . . . . . . 13 (∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜) ↔ (𝜑 → ∀𝑣𝑢 (𝑃𝑣) = 1𝑜))
1513, 14sylib 120 . . . . . . . . . . . 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 108 . . . . . . . . . . 11 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑃𝑢) = ∅)
1810, 11, 16, 17nninfalllemn 11542 . . . . . . . . . 10 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖𝑢, 1𝑜, ∅)))
1918fveq2d 5293 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄𝑃) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1𝑜, ∅))))
20 nninfalllem1.n0 . . . . . . . . . 10 (𝜑 → (𝑄𝑃) = ∅)
2120ad2antlr 473 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄𝑃) = ∅)
22 elequ2 1648 . . . . . . . . . . . . . 14 (𝑛 = 𝑢 → (𝑖𝑛𝑖𝑢))
2322ifbid 3408 . . . . . . . . . . . . 13 (𝑛 = 𝑢 → if(𝑖𝑛, 1𝑜, ∅) = if(𝑖𝑢, 1𝑜, ∅))
2423mpteq2dv 3921 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (𝑖 ∈ ω ↦ if(𝑖𝑛, 1𝑜, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑢, 1𝑜, ∅)))
2524fveq2d 5293 . . . . . . . . . . 11 (𝑛 = 𝑢 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1𝑜, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1𝑜, ∅))))
2625eqeq1d 2096 . . . . . . . . . 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 473 . . . . . . . . . 10 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1𝑜, ∅))) = 1𝑜)
2926, 28, 11rspcdva 2727 . . . . . . . . 9 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑢, 1𝑜, ∅))) = 1𝑜)
3019, 21, 293eqtr3d 2128 . . . . . . . 8 ((((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) ∧ (𝑃𝑢) = ∅) → ∅ = 1𝑜)
3130ex 113 . . . . . . 7 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → ((𝑃𝑢) = ∅ → ∅ = 1𝑜))
328, 31mtoi 625 . . . . . 6 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → ¬ (𝑃𝑢) = ∅)
33 fveq1 5288 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑃 → (𝑓‘suc 𝑗) = (𝑃‘suc 𝑗))
34 fveq1 5288 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑃 → (𝑓𝑗) = (𝑃𝑗))
3533, 34sseq12d 3053 . . . . . . . . . . . . . . 15 (𝑓 = 𝑃 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
3635ralbidv 2380 . . . . . . . . . . . . . 14 (𝑓 = 𝑃 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
37 df-nninf 6770 . . . . . . . . . . . . . 14 = {𝑓 ∈ (2𝑜𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
3836, 37elrab2 2772 . . . . . . . . . . . . 13 (𝑃 ∈ ℕ ↔ (𝑃 ∈ (2𝑜𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
399, 38sylib 120 . . . . . . . . . . . 12 (𝜑 → (𝑃 ∈ (2𝑜𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃𝑗)))
4039simpld 110 . . . . . . . . . . 11 (𝜑𝑃 ∈ (2𝑜𝑚 ω))
41 elmapi 6407 . . . . . . . . . . 11 (𝑃 ∈ (2𝑜𝑚 ω) → 𝑃:ω⟶2𝑜)
4240, 41syl 14 . . . . . . . . . 10 (𝜑𝑃:ω⟶2𝑜)
4342adantl 271 . . . . . . . . 9 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → 𝑃:ω⟶2𝑜)
44 simpll 496 . . . . . . . . 9 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → 𝑢 ∈ ω)
4543, 44ffvelrnd 5419 . . . . . . . 8 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → (𝑃𝑢) ∈ 2𝑜)
46 elpri 3464 . . . . . . . . 9 ((𝑃𝑢) ∈ {∅, 1𝑜} → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1𝑜))
47 df2o3 6177 . . . . . . . . 9 2𝑜 = {∅, 1𝑜}
4846, 47eleq2s 2182 . . . . . . . 8 ((𝑃𝑢) ∈ 2𝑜 → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1𝑜))
4945, 48syl 14 . . . . . . 7 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → ((𝑃𝑢) = ∅ ∨ (𝑃𝑢) = 1𝑜))
5049orcomd 683 . . . . . 6 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → ((𝑃𝑢) = 1𝑜 ∨ (𝑃𝑢) = ∅))
5132, 50ecased 1285 . . . . 5 (((𝑢 ∈ ω ∧ ∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜)) ∧ 𝜑) → (𝑃𝑢) = 1𝑜)
5251exp31 356 . . . 4 (𝑢 ∈ ω → (∀𝑣𝑢 (𝜑 → (𝑃𝑣) = 1𝑜) → (𝜑 → (𝑃𝑢) = 1𝑜)))
533, 6, 52omsinds 4425 . . 3 (𝑛 ∈ ω → (𝜑 → (𝑃𝑛) = 1𝑜))
5453impcom 123 . 2 ((𝜑𝑛 ∈ ω) → (𝑃𝑛) = 1𝑜)
5554ralrimiva 2446 1 (𝜑 → ∀𝑛 ∈ ω (𝑃𝑛) = 1𝑜)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 664   = wceq 1289  wcel 1438  wral 2359  wss 2997  c0 3284  ifcif 3389  {cpr 3442  cmpt 3891  suc csuc 4183  ωcom 4395  wf 4998  cfv 5002  (class class class)co 5634  1𝑜c1o 6156  2𝑜c2o 6157  𝑚 cmap 6385  xnninf 6768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770
This theorem is referenced by:  nninfall  11544
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