Theorem nninfalllem1 13888

Description: Lemma for nninfall 13889. (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 5486	. . . . . 6 ⊢ (𝑢 = 𝑣 → (𝑃‘𝑢) = (𝑃‘𝑣))
2	1	eqeq1d 2174	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝑃‘𝑢) = 1o ↔ (𝑃‘𝑣) = 1o))
3	2	imbi2d 229	. . . 4 ⊢ (𝑢 = 𝑣 → ((𝜑 → (𝑃‘𝑢) = 1o) ↔ (𝜑 → (𝑃‘𝑣) = 1o)))
4		fveq2 5486	. . . . . 6 ⊢ (𝑢 = 𝑛 → (𝑃‘𝑢) = (𝑃‘𝑛))
5	4	eqeq1d 2174	. . . . 5 ⊢ (𝑢 = 𝑛 → ((𝑃‘𝑢) = 1o ↔ (𝑃‘𝑛) = 1o))
6	5	imbi2d 229	. . . 4 ⊢ (𝑢 = 𝑛 → ((𝜑 → (𝑃‘𝑢) = 1o) ↔ (𝜑 → (𝑃‘𝑛) = 1o)))
7		1n0 6400	. . . . . . . 8 ⊢ 1o ≠ ∅
8	7	nesymi 2382	. . . . . . 7 ⊢ ¬ ∅ = 1o
9		nninfalllem1.p	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ∞)
10	9	ad2antlr 481	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝑃 ∈ ℕ∞)
11		simplll 523	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝑢 ∈ ω)
12		simplr 520	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝜑)
13		simpllr 524	. . . . . . . . . . . . 13 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o))
14		r19.21v 2543	. . . . . . . . . . . . 13 ⊢ (∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o) ↔ (𝜑 → ∀𝑣 ∈ 𝑢 (𝑃‘𝑣) = 1o))
15	13, 14	sylib 121	. . . . . . . . . . . 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 109	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → (𝑃‘𝑢) = ∅)
18	10, 11, 16, 17	nnnninfeq 7092	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅)))
19	18	fveq2d 5490	. . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → (𝑄‘𝑃) = ∅)
22		elequ2 2141	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑢 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 𝑢))
23	22	ifbid 3541	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑢 → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 𝑢, 1o, ∅))
24	23	mpteq2dv 4073	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅)))
25	24	fveq2d 5490	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑢 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅))))
26	25	eqeq1d 2174	. . . . . . . . . 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 481	. . . . . . . . . 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 2835	. . . . . . . . 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 2206	. . . . . . . 8 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → ∅ = 1o)
31	30	ex 114	. . . . . . 7 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ((𝑃‘𝑢) = ∅ → ∅ = 1o))
32	8, 31	mtoi 654	. . . . . 6 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ¬ (𝑃‘𝑢) = ∅)
33		fveq1 5485	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑃 → (𝑓‘suc 𝑗) = (𝑃‘suc 𝑗))
34		fveq1 5485	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑗) = (𝑃‘𝑗))
35	33, 34	sseq12d 3173	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑃 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
36	35	ralbidv 2466	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑃 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
37		df-nninf 7085	. . . . . . . . . . . . . 14 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
38	36, 37	elrab2 2885	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ∞ ↔ (𝑃 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
39	9, 38	sylib 121	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
40	39	simpld 111	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 ω))
41		elmapi 6636	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (2o ↑𝑚 ω) → 𝑃:ω⟶2o)
42	40, 41	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:ω⟶2o)
43	42	adantl 275	. . . . . . . . 9 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → 𝑃:ω⟶2o)
44		simpll 519	. . . . . . . . 9 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → 𝑢 ∈ ω)
45	43, 44	ffvelrnd 5621	. . . . . . . 8 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → (𝑃‘𝑢) ∈ 2o)
46		elpri 3599	. . . . . . . . 9 ⊢ ((𝑃‘𝑢) ∈ {∅, 1o} → ((𝑃‘𝑢) = ∅ ∨ (𝑃‘𝑢) = 1o))
47		df2o3 6398	. . . . . . . . 9 ⊢ 2o = {∅, 1o}
48	46, 47	eleq2s 2261	. . . . . . . 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 719	. . . . . 6 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ((𝑃‘𝑢) = 1o ∨ (𝑃‘𝑢) = ∅))
51	32, 50	ecased 1339	. . . . 5 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → (𝑃‘𝑢) = 1o)
52	51	exp31 362	. . . 4 ⊢ (𝑢 ∈ ω → (∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o) → (𝜑 → (𝑃‘𝑢) = 1o)))
53	3, 6, 52	omsinds 4599	. . 3 ⊢ (𝑛 ∈ ω → (𝜑 → (𝑃‘𝑛) = 1o))
54	53	impcom 124	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝑃‘𝑛) = 1o)
55	54	ralrimiva 2539	1 ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑃‘𝑛) = 1o)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  ∅c0 3409  ifcif 3520  {cpr 3577   ↦ cmpt 4043  suc csuc 4343  ωcom 4567  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  1_oc1o 6377  2_oc2o 6378   ↑_𝑚 cmap 6614  ℕ_∞xnninf 7084

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1o 6384  df-2o 6385  df-map 6616  df-nninf 7085

This theorem is referenced by:  nninfall  13889