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Mirrors > Home > ILE Home > Th. List > addpinq1 | GIF version |
Description: Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Ref | Expression |
---|---|
addpinq1 | ⊢ (𝐴 ∈ N → [〈(𝐴 +N 1o), 1o〉] ~Q = ([〈𝐴, 1o〉] ~Q +Q 1Q)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1nqqs 7183 | . . . . 5 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
2 | 1 | oveq2i 5793 | . . . 4 ⊢ ([〈𝐴, 1o〉] ~Q +Q 1Q) = ([〈𝐴, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) |
3 | 1pi 7147 | . . . . 5 ⊢ 1o ∈ N | |
4 | addpipqqs 7202 | . . . . . 6 ⊢ (((𝐴 ∈ N ∧ 1o ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → ([〈𝐴, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) = [〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ) | |
5 | 3, 3, 4 | mpanr12 436 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → ([〈𝐴, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) = [〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ) |
6 | 3, 5 | mpan2 422 | . . . 4 ⊢ (𝐴 ∈ N → ([〈𝐴, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) = [〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ) |
7 | 2, 6 | syl5eq 2185 | . . 3 ⊢ (𝐴 ∈ N → ([〈𝐴, 1o〉] ~Q +Q 1Q) = [〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ) |
8 | mulidpi 7150 | . . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o) | |
9 | 3, 8 | ax-mp 5 | . . . . . 6 ⊢ (1o ·N 1o) = 1o |
10 | 9 | oveq2i 5793 | . . . . 5 ⊢ ((𝐴 ·N 1o) +N (1o ·N 1o)) = ((𝐴 ·N 1o) +N 1o) |
11 | 10, 9 | opeq12i 3718 | . . . 4 ⊢ 〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉 = 〈((𝐴 ·N 1o) +N 1o), 1o〉 |
12 | eceq1 6472 | . . . 4 ⊢ (〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉 = 〈((𝐴 ·N 1o) +N 1o), 1o〉 → [〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q = [〈((𝐴 ·N 1o) +N 1o), 1o〉] ~Q ) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ [〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q = [〈((𝐴 ·N 1o) +N 1o), 1o〉] ~Q |
14 | 7, 13 | eqtrdi 2189 | . 2 ⊢ (𝐴 ∈ N → ([〈𝐴, 1o〉] ~Q +Q 1Q) = [〈((𝐴 ·N 1o) +N 1o), 1o〉] ~Q ) |
15 | mulidpi 7150 | . . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
16 | 15 | oveq1d 5797 | . . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1o) +N 1o) = (𝐴 +N 1o)) |
17 | 16 | opeq1d 3719 | . . 3 ⊢ (𝐴 ∈ N → 〈((𝐴 ·N 1o) +N 1o), 1o〉 = 〈(𝐴 +N 1o), 1o〉) |
18 | 17 | eceq1d 6473 | . 2 ⊢ (𝐴 ∈ N → [〈((𝐴 ·N 1o) +N 1o), 1o〉] ~Q = [〈(𝐴 +N 1o), 1o〉] ~Q ) |
19 | 14, 18 | eqtr2d 2174 | 1 ⊢ (𝐴 ∈ N → [〈(𝐴 +N 1o), 1o〉] ~Q = ([〈𝐴, 1o〉] ~Q +Q 1Q)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 〈cop 3535 (class class class)co 5782 1oc1o 6314 [cec 6435 Ncnpi 7104 +N cpli 7105 ·N cmi 7106 ~Q ceq 7111 1Qc1q 7113 +Q cplq 7114 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-plpq 7176 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-1nqqs 7183 |
This theorem is referenced by: pitonnlem2 7679 |
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