Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > archiexdiv | Structured version Visualization version GIF version |
Description: In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
Ref | Expression |
---|---|
archiexdiv.b | ⊢ 𝐵 = (Base‘𝑊) |
archiexdiv.0 | ⊢ 0 = (0g‘𝑊) |
archiexdiv.i | ⊢ < = (lt‘𝑊) |
archiexdiv.x | ⊢ · = (.g‘𝑊) |
Ref | Expression |
---|---|
archiexdiv | ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | archiexdiv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
2 | archiexdiv.0 | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
3 | archiexdiv.i | . . . . 5 ⊢ < = (lt‘𝑊) | |
4 | archiexdiv.x | . . . . 5 ⊢ · = (.g‘𝑊) | |
5 | 1, 2, 3, 4 | isarchi3 30816 | . . . 4 ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)))) |
6 | 5 | biimpa 479 | . . 3 ⊢ ((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))) |
7 | 6 | 3ad2ant1 1129 | . 2 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))) |
8 | simp3 1134 | . 2 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → 0 < 𝑋) | |
9 | breq2 5070 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 < 𝑥 ↔ 0 < 𝑋)) | |
10 | oveq2 7164 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑛 · 𝑥) = (𝑛 · 𝑋)) | |
11 | 10 | breq2d 5078 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 < (𝑛 · 𝑥) ↔ 𝑦 < (𝑛 · 𝑋))) |
12 | 11 | rexbidv 3297 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋))) |
13 | 9, 12 | imbi12d 347 | . . . 4 ⊢ (𝑥 = 𝑋 → (( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)) ↔ ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋)))) |
14 | breq1 5069 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 < (𝑛 · 𝑋) ↔ 𝑌 < (𝑛 · 𝑋))) | |
15 | 14 | rexbidv 3297 | . . . . 5 ⊢ (𝑦 = 𝑌 → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋) ↔ ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋))) |
16 | 15 | imbi2d 343 | . . . 4 ⊢ (𝑦 = 𝑌 → (( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋)) ↔ ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)))) |
17 | 13, 16 | rspc2v 3633 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)) → ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)))) |
18 | 17 | 3ad2ant2 1130 | . 2 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)) → ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)))) |
19 | 7, 8, 18 | mp2d 49 | 1 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℕcn 11638 Basecbs 16483 0gc0g 16713 ltcplt 17551 .gcmg 18224 oGrpcogrp 30699 Archicarchi 30806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-toset 17644 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mulg 18225 df-omnd 30700 df-ogrp 30701 df-inftm 30807 df-archi 30808 |
This theorem is referenced by: (None) |
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