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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 7595. (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 7400 | . . 3 ⊢ 1Q ∈ Q | |
2 | addnqpr 7595 | . . 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 425 | . 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 7501 | . . 3 ⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | |
5 | 4 | oveq2i 5911 | . 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 | eqtr4di 2240 | 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 1364 ∈ wcel 2160 {cab 2175 〈cop 3613 class class class wbr 4021 (class class class)co 5900 Qcnq 7314 1Qc1q 7315 +Q cplq 7316 <Q cltq 7319 1Pc1p 7326 +P cpp 7327 |
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-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-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 |
This theorem is referenced by: pitonnlem2 7881 |
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