Theorem nninfalllem1 15498

Description: Lemma for nninfall 15499. (Contributed by Jim Kingdon, 1-Aug-2022.)

Hypotheses

Ref	Expression
nninfall.q	⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
nninfall.inf	⊢ (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
nninfall.n	⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
nninfalllem1.p	⊢ (𝜑 → 𝑃 ∈ ℕ∞)
nninfalllem1.n0	⊢ (𝜑 → (𝑄‘𝑃) = ∅)

Assertion

Ref	Expression
nninfalllem1	⊢ (𝜑 → ∀𝑛 ∈ ω (𝑃‘𝑛) = 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.

Step	Hyp	Ref	Expression
1		fveq2 5554	. . . . . 6 ⊢ (𝑢 = 𝑣 → (𝑃‘𝑢) = (𝑃‘𝑣))
2	1	eqeq1d 2202	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝑃‘𝑢) = 1o ↔ (𝑃‘𝑣) = 1o))
3	2	imbi2d 230	. . . 4 ⊢ (𝑢 = 𝑣 → ((𝜑 → (𝑃‘𝑢) = 1o) ↔ (𝜑 → (𝑃‘𝑣) = 1o)))
4		fveq2 5554	. . . . . 6 ⊢ (𝑢 = 𝑛 → (𝑃‘𝑢) = (𝑃‘𝑛))
5	4	eqeq1d 2202	. . . . 5 ⊢ (𝑢 = 𝑛 → ((𝑃‘𝑢) = 1o ↔ (𝑃‘𝑛) = 1o))
6	5	imbi2d 230	. . . 4 ⊢ (𝑢 = 𝑛 → ((𝜑 → (𝑃‘𝑢) = 1o) ↔ (𝜑 → (𝑃‘𝑛) = 1o)))
7		1n0 6485	. . . . . . . 8 ⊢ 1o ≠ ∅
8	7	nesymi 2410	. . . . . . 7 ⊢ ¬ ∅ = 1o
9		nninfalllem1.p	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ∞)
10	9	ad2antlr 489	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝑃 ∈ ℕ∞)
11		simplll 533	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝑢 ∈ ω)
12		simplr 528	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝜑)
13		simpllr 534	. . . . . . . . . . . . 13 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o))
14		r19.21v 2571	. . . . . . . . . . . . 13 ⊢ (∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o) ↔ (𝜑 → ∀𝑣 ∈ 𝑢 (𝑃‘𝑣) = 1o))
15	13, 14	sylib 122	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → (𝜑 → ∀𝑣 ∈ 𝑢 (𝑃‘𝑣) = 1o))
16	12, 15	mpd 13	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → ∀𝑣 ∈ 𝑢 (𝑃‘𝑣) = 1o)
17		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → (𝑃‘𝑢) = ∅)
18	10, 11, 16, 17	nnnninfeq 7187	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅)))
19	18	fveq2d 5558	. . . . . . . . 9 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → (𝑄‘𝑃) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅))))
20		nninfalllem1.n0	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘𝑃) = ∅)
21	20	ad2antlr 489	. . . . . . . . 9 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → (𝑄‘𝑃) = ∅)
22		elequ2 2169	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑢 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 𝑢))
23	22	ifbid 3578	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑢 → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 𝑢, 1o, ∅))
24	23	mpteq2dv 4120	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅)))
25	24	fveq2d 5558	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑢 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅))))
26	25	eqeq1d 2202	. . . . . . . . . 10 ⊢ (𝑛 = 𝑢 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅))) = 1o))
27		nninfall.n	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
28	27	ad2antlr 489	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
29	26, 28, 11	rspcdva 2869	. . . . . . . . 9 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅))) = 1o)
30	19, 21, 29	3eqtr3d 2234	. . . . . . . 8 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → ∅ = 1o)
31	30	ex 115	. . . . . . 7 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ((𝑃‘𝑢) = ∅ → ∅ = 1o))
32	8, 31	mtoi 665	. . . . . 6 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ¬ (𝑃‘𝑢) = ∅)
33		fveq1 5553	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑃 → (𝑓‘suc 𝑗) = (𝑃‘suc 𝑗))
34		fveq1 5553	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑗) = (𝑃‘𝑗))
35	33, 34	sseq12d 3210	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑃 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
36	35	ralbidv 2494	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑃 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
37		df-nninf 7179	. . . . . . . . . . . . . 14 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
38	36, 37	elrab2 2919	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ∞ ↔ (𝑃 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
39	9, 38	sylib 122	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
40	39	simpld 112	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 ω))
41		elmapi 6724	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (2o ↑𝑚 ω) → 𝑃:ω⟶2o)
42	40, 41	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:ω⟶2o)
43	42	adantl 277	. . . . . . . . 9 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → 𝑃:ω⟶2o)
44		simpll 527	. . . . . . . . 9 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → 𝑢 ∈ ω)
45	43, 44	ffvelcdmd 5694	. . . . . . . 8 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → (𝑃‘𝑢) ∈ 2o)
46		elpri 3641	. . . . . . . . 9 ⊢ ((𝑃‘𝑢) ∈ {∅, 1o} → ((𝑃‘𝑢) = ∅ ∨ (𝑃‘𝑢) = 1o))
47		df2o3 6483	. . . . . . . . 9 ⊢ 2o = {∅, 1o}
48	46, 47	eleq2s 2288	. . . . . . . 8 ⊢ ((𝑃‘𝑢) ∈ 2o → ((𝑃‘𝑢) = ∅ ∨ (𝑃‘𝑢) = 1o))
49	45, 48	syl 14	. . . . . . 7 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ((𝑃‘𝑢) = ∅ ∨ (𝑃‘𝑢) = 1o))
50	49	orcomd 730	. . . . . 6 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ((𝑃‘𝑢) = 1o ∨ (𝑃‘𝑢) = ∅))
51	32, 50	ecased 1360	. . . . 5 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → (𝑃‘𝑢) = 1o)
52	51	exp31 364	. . . 4 ⊢ (𝑢 ∈ ω → (∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o) → (𝜑 → (𝑃‘𝑢) = 1o)))
53	3, 6, 52	omsinds 4654	. . 3 ⊢ (𝑛 ∈ ω → (𝜑 → (𝑃‘𝑛) = 1o))
54	53	impcom 125	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝑃‘𝑛) = 1o)
55	54	ralrimiva 2567	1 ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑃‘𝑛) = 1o)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153  ∅c0 3446  ifcif 3557  {cpr 3619   ↦ cmpt 4090  suc csuc 4396  ωcom 4622  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  1_oc1o 6462  2_oc2o 6463   ↑_𝑚 cmap 6702  ℕ_∞xnninf 7178

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1o 6469  df-2o 6470  df-map 6704  df-nninf 7179

This theorem is referenced by:  nninfall  15499