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Mirrors > Home > ILE Home > Th. List > nninfdclemf1 | GIF version |
Description: Lemma for nninfdc 12280. The function from nninfdclemf 12276 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.) |
Ref | Expression |
---|---|
nninfdclemf.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
nninfdclemf.dc | ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) |
nninfdclemf.nb | ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) |
nninfdclemf.j | ⊢ (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽)) |
nninfdclemf.f | ⊢ 𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽)) |
Ref | Expression |
---|---|
nninfdclemf1 | ⊢ (𝜑 → 𝐹:ℕ–1-1→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfdclemf.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
2 | nninfdclemf.dc | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) | |
3 | nninfdclemf.nb | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) | |
4 | nninfdclemf.j | . . 3 ⊢ (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽)) | |
5 | nninfdclemf.f | . . 3 ⊢ 𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽)) | |
6 | 1, 2, 3, 4, 5 | nninfdclemf 12276 | . 2 ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) |
7 | fveq2 5471 | . . . . 5 ⊢ (𝑢 = 𝑣 → (𝐹‘𝑢) = (𝐹‘𝑣)) | |
8 | fveq2 5471 | . . . . 5 ⊢ (𝑢 = 𝑝 → (𝐹‘𝑢) = (𝐹‘𝑝)) | |
9 | fveq2 5471 | . . . . 5 ⊢ (𝑢 = 𝑞 → (𝐹‘𝑢) = (𝐹‘𝑞)) | |
10 | nnssre 8843 | . . . . 5 ⊢ ℕ ⊆ ℝ | |
11 | 1 | adantr 274 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ℕ) → 𝐴 ⊆ ℕ) |
12 | 6 | ffvelrnda 5605 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ℕ) → (𝐹‘𝑢) ∈ 𝐴) |
13 | 11, 12 | sseldd 3129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ℕ) → (𝐹‘𝑢) ∈ ℕ) |
14 | 13 | nnred 8852 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ℕ) → (𝐹‘𝑢) ∈ ℝ) |
15 | 1 | ad2antrr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → 𝐴 ⊆ ℕ) |
16 | 2 | ad2antrr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) |
17 | 3 | ad2antrr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) |
18 | 4 | ad2antrr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽)) |
19 | simplrl 525 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → 𝑢 ∈ ℕ) | |
20 | simplrr 526 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → 𝑣 ∈ ℕ) | |
21 | simpr 109 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → 𝑢 < 𝑣) | |
22 | 15, 16, 17, 18, 5, 19, 20, 21 | nninfdclemlt 12278 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) ∧ 𝑢 < 𝑣) → (𝐹‘𝑢) < (𝐹‘𝑣)) |
23 | 22 | ex 114 | . . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ ℕ ∧ 𝑣 ∈ ℕ)) → (𝑢 < 𝑣 → (𝐹‘𝑢) < (𝐹‘𝑣))) |
24 | 7, 8, 9, 10, 14, 23 | eqord1 8363 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → (𝑝 = 𝑞 ↔ (𝐹‘𝑝) = (𝐹‘𝑞))) |
25 | 24 | biimprd 157 | . . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞)) |
26 | 25 | ralrimivva 2539 | . 2 ⊢ (𝜑 → ∀𝑝 ∈ ℕ ∀𝑞 ∈ ℕ ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞)) |
27 | dff13 5721 | . 2 ⊢ (𝐹:ℕ–1-1→𝐴 ↔ (𝐹:ℕ⟶𝐴 ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ ℕ ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))) | |
28 | 6, 26, 27 | sylanbrc 414 | 1 ⊢ (𝜑 → 𝐹:ℕ–1-1→𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ∃wrex 2436 ∩ cin 3101 ⊆ wss 3102 class class class wbr 3967 ↦ cmpt 4028 ⟶wf 5169 –1-1→wf1 5170 ‘cfv 5173 (class class class)co 5827 ∈ cmpo 5829 infcinf 6930 ℝcr 7734 1c1 7736 + caddc 7738 < clt 7915 ℕcn 8839 ℤ≥cuz 9445 seqcseq 10354 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-isom 5182 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-frec 6341 df-sup 6931 df-inf 6932 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-inn 8840 df-n0 9097 df-z 9174 df-uz 9446 df-fz 9920 df-fzo 10052 df-seqfrec 10355 |
This theorem is referenced by: nninfdc 12280 |
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