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Mirrors > Home > ILE Home > Th. List > addnqpr1 | GIF version |
Description: Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 7118. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Ref | Expression |
---|---|
addnqpr1 | ⊢ (𝐴 ∈ Q → 〈{𝑙 ∣ 𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}〉 = (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 +P 1P)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nq 6923 | . . 3 ⊢ 1Q ∈ Q | |
2 | addnqpr 7118 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → 〈{𝑙 ∣ 𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}〉 = (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 +P 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉)) | |
3 | 1, 2 | mpan2 416 | . 2 ⊢ (𝐴 ∈ Q → 〈{𝑙 ∣ 𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}〉 = (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 +P 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉)) |
4 | df-i1p 7024 | . . 3 ⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | |
5 | 4 | oveq2i 5663 | . 2 ⊢ (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 +P 1P) = (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 +P 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉) |
6 | 3, 5 | syl6eqr 2138 | 1 ⊢ (𝐴 ∈ Q → 〈{𝑙 ∣ 𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}〉 = (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 +P 1P)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ∈ wcel 1438 {cab 2074 〈cop 3449 class class class wbr 3845 (class class class)co 5652 Qcnq 6837 1Qc1q 6838 +Q cplq 6839 <Q cltq 6842 1Pc1p 6849 +P cpp 6850 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-eprel 4116 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-1o 6181 df-2o 6182 df-oadd 6185 df-omul 6186 df-er 6290 df-ec 6292 df-qs 6296 df-ni 6861 df-pli 6862 df-mi 6863 df-lti 6864 df-plpq 6901 df-mpq 6902 df-enq 6904 df-nqqs 6905 df-plqqs 6906 df-mqqs 6907 df-1nqqs 6908 df-rq 6909 df-ltnqqs 6910 df-enq0 6981 df-nq0 6982 df-0nq0 6983 df-plq0 6984 df-mq0 6985 df-inp 7023 df-i1p 7024 df-iplp 7025 |
This theorem is referenced by: pitonnlem2 7382 |
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