![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > q1mod | GIF version |
Description: Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Jim Kingdon, 21-Oct-2021.) |
Ref | Expression |
---|---|
q1mod | ⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 8874 | . . 3 ⊢ 1 ∈ ℤ | |
2 | zq 9210 | . . 3 ⊢ (1 ∈ ℤ → 1 ∈ ℚ) | |
3 | 1, 2 | mp1i 10 | . 2 ⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → 1 ∈ ℚ) |
4 | simpl 108 | . 2 ⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → 𝑁 ∈ ℚ) | |
5 | 0le1 8056 | . . 3 ⊢ 0 ≤ 1 | |
6 | 5 | a1i 9 | . 2 ⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → 0 ≤ 1) |
7 | simpr 109 | . 2 ⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → 1 < 𝑁) | |
8 | modqid 9905 | . 2 ⊢ (((1 ∈ ℚ ∧ 𝑁 ∈ ℚ) ∧ (0 ≤ 1 ∧ 1 < 𝑁)) → (1 mod 𝑁) = 1) | |
9 | 3, 4, 6, 7, 8 | syl22anc 1182 | 1 ⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 0cc0 7447 1c1 7448 < clt 7619 ≤ cle 7620 ℤcz 8848 ℚcq 9203 mod cmo 9878 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-n0 8772 df-z 8849 df-q 9204 df-rp 9234 df-fl 9826 df-mod 9879 |
This theorem is referenced by: mulp1mod1 9921 |
Copyright terms: Public domain | W3C validator |