Theorem nninfalllem1 14327

Description: Lemma for nninfall 14328. (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 5507	. . . . . 6 ⊢ (𝑢 = 𝑣 → (𝑃‘𝑢) = (𝑃‘𝑣))
2	1	eqeq1d 2184	. . . . 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 5507	. . . . . 6 ⊢ (𝑢 = 𝑛 → (𝑃‘𝑢) = (𝑃‘𝑛))
5	4	eqeq1d 2184	. . . . 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 6423	. . . . . . . 8 ⊢ 1o ≠ ∅
8	7	nesymi 2391	. . . . . . 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 2552	. . . . . . . . . . . . 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 7116	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) ∧ (𝑃‘𝑢) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅)))
19	18	fveq2d 5511	. . . . . . . . 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 2151	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑢 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 𝑢))
23	22	ifbid 3553	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑢 → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 𝑢, 1o, ∅))
24	23	mpteq2dv 4089	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅)))
25	24	fveq2d 5511	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑢 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑢, 1o, ∅))))
26	25	eqeq1d 2184	. . . . . . . . . 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 2844	. . . . . . . . 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 2216	. . . . . . . 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 664	. . . . . 6 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ¬ (𝑃‘𝑢) = ∅)
33		fveq1 5506	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑃 → (𝑓‘suc 𝑗) = (𝑃‘suc 𝑗))
34		fveq1 5506	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑗) = (𝑃‘𝑗))
35	33, 34	sseq12d 3184	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑃 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
36	35	ralbidv 2475	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑃 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑃‘suc 𝑗) ⊆ (𝑃‘𝑗)))
37		df-nninf 7109	. . . . . . . . . . . . . 14 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
38	36, 37	elrab2 2894	. . . . . . . . . . . . 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 6660	. . . . . . . . . . 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 5644	. . . . . . . 8 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → (𝑃‘𝑢) ∈ 2o)
46		elpri 3612	. . . . . . . . 9 ⊢ ((𝑃‘𝑢) ∈ {∅, 1o} → ((𝑃‘𝑢) = ∅ ∨ (𝑃‘𝑢) = 1o))
47		df2o3 6421	. . . . . . . . 9 ⊢ 2o = {∅, 1o}
48	46, 47	eleq2s 2270	. . . . . . . 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 729	. . . . . 6 ⊢ (((𝑢 ∈ ω ∧ ∀𝑣 ∈ 𝑢 (𝜑 → (𝑃‘𝑣) = 1o)) ∧ 𝜑) → ((𝑃‘𝑢) = 1o ∨ (𝑃‘𝑢) = ∅))
51	32, 50	ecased 1349	. . . . 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 4615	. . 3 ⊢ (𝑛 ∈ ω → (𝜑 → (𝑃‘𝑛) = 1o))
54	53	impcom 125	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝑃‘𝑛) = 1o)
55	54	ralrimiva 2548	1 ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑃‘𝑛) = 1o)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2146  ∀wral 2453   ⊆ wss 3127  ∅c0 3420  ifcif 3532  {cpr 3590   ↦ cmpt 4059  suc csuc 4359  ωcom 4583  ⟶wf 5204  ‘cfv 5208  (class class class)co 5865  1_oc1o 6400  2_oc2o 6401   ↑_𝑚 cmap 6638  ℕ_∞xnninf 7108

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1o 6407  df-2o 6408  df-map 6640  df-nninf 7109

This theorem is referenced by:  nninfall  14328