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Mirrors > Home > ILE Home > Th. List > axcaucvglemcl | GIF version |
Description: Lemma for axcaucvg 7877. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.) |
Ref | Expression |
---|---|
axcaucvglemcl.n | ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
axcaucvglemcl.f | ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) |
Ref | Expression |
---|---|
axcaucvglemcl | ⊢ ((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) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pitonn 7825 | . . . . . 6 ⊢ (𝐽 ∈ N → 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) | |
2 | axcaucvglemcl.n | . . . . . 6 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
3 | 1, 2 | eleqtrrdi 2271 | . . . . 5 ⊢ (𝐽 ∈ N → 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ 𝑁) |
4 | axcaucvglemcl.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) | |
5 | 4 | ffvelcdmda 5646 | . . . . 5 ⊢ ((𝜑 ∧ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ 𝑁) → (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) ∈ ℝ) |
6 | 3, 5 | sylan2 286 | . . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) ∈ ℝ) |
7 | elrealeu 7806 | . . . 4 ⊢ ((𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) ∈ ℝ ↔ ∃!𝑧 ∈ R 〈𝑧, 0R〉 = (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉)) | |
8 | 6, 7 | sylib 122 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ∃!𝑧 ∈ R 〈𝑧, 0R〉 = (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉)) |
9 | eqcom 2179 | . . . 4 ⊢ (〈𝑧, 0R〉 = (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) ↔ (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 〈𝑧, 0R〉) | |
10 | 9 | reubii 2662 | . . 3 ⊢ (∃!𝑧 ∈ R 〈𝑧, 0R〉 = (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) ↔ ∃!𝑧 ∈ R (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 〈𝑧, 0R〉) |
11 | 8, 10 | sylib 122 | . 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〉) |
12 | riotacl 5838 | . 2 ⊢ (∃!𝑧 ∈ R (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 〈𝑧, 0R〉 → (℩𝑧 ∈ R (𝐹‘〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 〈𝑧, 0R〉) ∈ R) | |
13 | 11, 12 | syl 14 | 1 ⊢ ((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) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {cab 2163 ∀wral 2455 ∃!wreu 2457 〈cop 3594 ∩ cint 3842 class class class wbr 4000 ⟶wf 5207 ‘cfv 5211 ℩crio 5823 (class class class)co 5868 1oc1o 6403 [cec 6526 Ncnpi 7249 ~Q ceq 7256 <Q cltq 7262 1Pc1p 7269 +P cpp 7270 ~R cer 7273 Rcnr 7274 0Rc0r 7275 ℝcr 7788 1c1 7790 + caddc 7792 |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
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 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4285 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-irdg 6364 df-1o 6410 df-2o 6411 df-oadd 6414 df-omul 6415 df-er 6528 df-ec 6530 df-qs 6534 df-ni 7281 df-pli 7282 df-mi 7283 df-lti 7284 df-plpq 7321 df-mpq 7322 df-enq 7324 df-nqqs 7325 df-plqqs 7326 df-mqqs 7327 df-1nqqs 7328 df-rq 7329 df-ltnqqs 7330 df-enq0 7401 df-nq0 7402 df-0nq0 7403 df-plq0 7404 df-mq0 7405 df-inp 7443 df-i1p 7444 df-iplp 7445 df-enr 7703 df-nr 7704 df-plr 7705 df-0r 7708 df-1r 7709 df-c 7795 df-1 7797 df-r 7799 df-add 7800 |
This theorem is referenced by: axcaucvglemf 7873 axcaucvglemval 7874 |
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