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Theorem flqge 10351

Description: The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)

**Assertion**
Ref	Expression
flqge	⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴)))

**Proof of Theorem flqge**
Step	Hyp	Ref	Expression
1		flqltp1 10348	. . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))
2	1	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐴 < ((⌊‘𝐴) + 1))
3		simpr 110	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
4	3	zred 9439	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ)
5		qre 9690	. . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
6	5	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ)
7		simpl 109	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℚ)
8	7	flqcld 10346	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ)
9	8	peano2zd 9442	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℤ)
10	9	zred 9439	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ)
11		lelttr 8108	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((⌊‘𝐴) + 1) ∈ ℝ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)) → 𝐵 < ((⌊‘𝐴) + 1)))
12	4, 6, 10, 11	syl3anc 1249	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)) → 𝐵 < ((⌊‘𝐴) + 1)))
13	2, 12	mpan2d 428	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 → 𝐵 < ((⌊‘𝐴) + 1)))
14		zleltp1 9372	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ (⌊‘𝐴) ∈ ℤ) → (𝐵 ≤ (⌊‘𝐴) ↔ 𝐵 < ((⌊‘𝐴) + 1)))
15	3, 8, 14	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ (⌊‘𝐴) ↔ 𝐵 < ((⌊‘𝐴) + 1)))
16	13, 15	sylibrd 169	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 → 𝐵 ≤ (⌊‘𝐴)))
17		flqle 10347	. . . 4 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
18	17	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴)
19	8	zred 9439	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ)
20		letr 8102	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ (⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ (⌊‘𝐴) ∧ (⌊‘𝐴) ≤ 𝐴) → 𝐵 ≤ 𝐴))
21	4, 19, 6, 20	syl3anc 1249	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ (⌊‘𝐴) ∧ (⌊‘𝐴) ≤ 𝐴) → 𝐵 ≤ 𝐴))
22	18, 21	mpan2d 428	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ (⌊‘𝐴) → 𝐵 ≤ 𝐴))
23	16, 22	impbid 129	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2164 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 ℝcr 7871 1c1 7873 + caddc 7875 < clt 8054 ≤ cle 8055 ℤcz 9317 ℚcq 9684 ⌊cfl 10337

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 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-q 9685 df-rp 9720 df-fl 10339

This theorem is referenced by: flqlt 10352 flid 10353 flqwordi 10357 flqge0nn0 10362 flqge1nn 10363 flqmulnn0 10368 modqmuladdnn0 10439 hashdvds 12359 gausslemma2dlem4 15180 lgsquadlem1 15191

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