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Mirrors > Home > ILE Home > Th. List > axcaucvglemf | GIF version |
Description: Lemma for axcaucvg 7934. Mapping to N and R yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.) |
Ref | Expression |
---|---|
axcaucvg.n | ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
axcaucvg.f | ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) |
axcaucvg.cau | ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))) |
axcaucvg.g | ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑗, 1o〉] ~Q }, {𝑢 ∣ [〈𝑗, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 〈𝑧, 0R〉)) |
Ref | Expression |
---|---|
axcaucvglemf | ⊢ (𝜑 → 𝐺:N⟶R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axcaucvg.n | . . 3 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
2 | axcaucvg.f | . . 3 ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) | |
3 | 1, 2 | axcaucvglemcl 7929 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ N) → (℩𝑧 ∈ R (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑗, 1o〉] ~Q }, {𝑢 ∣ [〈𝑗, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 〈𝑧, 0R〉) ∈ R) |
4 | axcaucvg.g | . 2 ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑗, 1o〉] ~Q }, {𝑢 ∣ [〈𝑗, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 〈𝑧, 0R〉)) | |
5 | 3, 4 | fmptd 5694 | 1 ⊢ (𝜑 → 𝐺:N⟶R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 {cab 2175 ∀wral 2468 〈cop 3613 ∩ cint 3862 class class class wbr 4021 ↦ cmpt 4082 ⟶wf 5234 ‘cfv 5238 ℩crio 5854 (class class class)co 5900 1oc1o 6438 [cec 6561 Ncnpi 7306 ~Q ceq 7313 <Q cltq 7319 1Pc1p 7326 +P cpp 7327 ~R cer 7330 Rcnr 7331 0Rc0r 7332 ℝcr 7845 1c1 7847 + caddc 7849 <ℝ cltrr 7850 · cmul 7851 |
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 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-eprel 4310 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-irdg 6399 df-1o 6445 df-2o 6446 df-oadd 6449 df-omul 6450 df-er 6563 df-ec 6565 df-qs 6569 df-ni 7338 df-pli 7339 df-mi 7340 df-lti 7341 df-plpq 7378 df-mpq 7379 df-enq 7381 df-nqqs 7382 df-plqqs 7383 df-mqqs 7384 df-1nqqs 7385 df-rq 7386 df-ltnqqs 7387 df-enq0 7458 df-nq0 7459 df-0nq0 7460 df-plq0 7461 df-mq0 7462 df-inp 7500 df-i1p 7501 df-iplp 7502 df-enr 7760 df-nr 7761 df-plr 7762 df-0r 7765 df-1r 7766 df-c 7852 df-1 7854 df-r 7856 df-add 7857 |
This theorem is referenced by: axcaucvglemcau 7932 axcaucvglemres 7933 |
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