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Theorem flqge 10497

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 10494	. . . . 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 9565	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ)
5		qre 9816	. . . . . 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 10492	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ)
9	8	peano2zd 9568	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℤ)
10	9	zred 9565	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ)
11		lelttr 8231	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((⌊‘𝐴) + 1) ∈ ℝ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)) → 𝐵 < ((⌊‘𝐴) + 1)))
12	4, 6, 10, 11	syl3anc 1271	. . . 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 9498	. . . 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 10493	. . . 4 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
18	17	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴)
19	8	zred 9565	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ)
20		letr 8225	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ (⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ (⌊‘𝐴) ∧ (⌊‘𝐴) ≤ 𝐴) → 𝐵 ≤ 𝐴))
21	4, 19, 6, 20	syl3anc 1271	. . 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 2200 class class class wbr 4082 ‘cfv 5317 (class class class)co 6000 ℝcr 7994 1c1 7996 + caddc 7998 < clt 8177 ≤ cle 8178 ℤcz 9442 ℚcq 9810 ⌊cfl 10483

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-po 4386 df-iso 4387 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-n0 9366 df-z 9443 df-q 9811 df-rp 9846 df-fl 10485

This theorem is referenced by: flqlt 10498 flid 10499 flqwordi 10503 flqge0nn0 10508 flqge1nn 10509 flqmulnn0 10514 modqmuladdnn0 10585 hashdvds 12738 gausslemma2dlem4 15737 lgsquadlem1 15750 lgsquadlem2 15751

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