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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 7152 | . . . . 5 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
2 | 1 | oveq2i 5778 | . . . 4 ⊢ ([〈𝐴, 1o〉] ~Q +Q 1Q) = ([〈𝐴, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) |
3 | 1pi 7116 | . . . . 5 ⊢ 1o ∈ N | |
4 | addpipqqs 7171 | . . . . . 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 435 | . . . . 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 421 | . . . 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 2182 | . . 3 ⊢ (𝐴 ∈ N → ([〈𝐴, 1o〉] ~Q +Q 1Q) = [〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ) |
8 | mulidpi 7119 | . . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o) | |
9 | 3, 8 | ax-mp 5 | . . . . . 6 ⊢ (1o ·N 1o) = 1o |
10 | 9 | oveq2i 5778 | . . . . 5 ⊢ ((𝐴 ·N 1o) +N (1o ·N 1o)) = ((𝐴 ·N 1o) +N 1o) |
11 | 10, 9 | opeq12i 3705 | . . . 4 ⊢ 〈((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉 = 〈((𝐴 ·N 1o) +N 1o), 1o〉 |
12 | eceq1 6457 | . . . 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 | syl6eq 2186 | . 2 ⊢ (𝐴 ∈ N → ([〈𝐴, 1o〉] ~Q +Q 1Q) = [〈((𝐴 ·N 1o) +N 1o), 1o〉] ~Q ) |
15 | mulidpi 7119 | . . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
16 | 15 | oveq1d 5782 | . . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1o) +N 1o) = (𝐴 +N 1o)) |
17 | 16 | opeq1d 3706 | . . 3 ⊢ (𝐴 ∈ N → 〈((𝐴 ·N 1o) +N 1o), 1o〉 = 〈(𝐴 +N 1o), 1o〉) |
18 | 17 | eceq1d 6458 | . 2 ⊢ (𝐴 ∈ N → [〈((𝐴 ·N 1o) +N 1o), 1o〉] ~Q = [〈(𝐴 +N 1o), 1o〉] ~Q ) |
19 | 14, 18 | eqtr2d 2171 | 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 1331 ∈ wcel 1480 〈cop 3525 (class class class)co 5767 1oc1o 6299 [cec 6420 Ncnpi 7073 +N cpli 7074 ·N cmi 7075 ~Q ceq 7080 1Qc1q 7082 +Q cplq 7083 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-plpq 7145 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-1nqqs 7152 |
This theorem is referenced by: pitonnlem2 7648 |
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