Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omndmul3 | Structured version Visualization version GIF version |
Description: In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
omndmul.0 | ⊢ 𝐵 = (Base‘𝑀) |
omndmul.1 | ⊢ ≤ = (le‘𝑀) |
omndmul3.m | ⊢ · = (.g‘𝑀) |
omndmul3.0 | ⊢ 0 = (0g‘𝑀) |
omndmul3.o | ⊢ (𝜑 → 𝑀 ∈ oMnd) |
omndmul3.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
omndmul3.2 | ⊢ (𝜑 → 𝑃 ∈ ℕ0) |
omndmul3.3 | ⊢ (𝜑 → 𝑁 ≤ 𝑃) |
omndmul3.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
omndmul3.5 | ⊢ (𝜑 → 0 ≤ 𝑋) |
Ref | Expression |
---|---|
omndmul3 | ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omndmul3.o | . . 3 ⊢ (𝜑 → 𝑀 ∈ oMnd) | |
2 | omndmnd 30705 | . . . . 5 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd) |
4 | omndmul.0 | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
5 | omndmul3.0 | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
6 | 4, 5 | mndidcl 17926 | . . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵) |
7 | 3, 6 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) |
8 | omndmul3.1 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
9 | omndmul3.2 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ0) | |
10 | omndmul3.3 | . . . . 5 ⊢ (𝜑 → 𝑁 ≤ 𝑃) | |
11 | nn0sub 11948 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) → (𝑁 ≤ 𝑃 ↔ (𝑃 − 𝑁) ∈ ℕ0)) | |
12 | 11 | biimpa 479 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) ∧ 𝑁 ≤ 𝑃) → (𝑃 − 𝑁) ∈ ℕ0) |
13 | 8, 9, 10, 12 | syl21anc 835 | . . . 4 ⊢ (𝜑 → (𝑃 − 𝑁) ∈ ℕ0) |
14 | omndmul3.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | omndmul3.m | . . . . 5 ⊢ · = (.g‘𝑀) | |
16 | 4, 15 | mulgnn0cl 18244 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ (𝑃 − 𝑁) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵) |
17 | 3, 13, 14, 16 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵) |
18 | 4, 15 | mulgnn0cl 18244 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
19 | 3, 8, 14, 18 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) |
20 | omndmul3.5 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝑋) | |
21 | omndmul.1 | . . . . 5 ⊢ ≤ = (le‘𝑀) | |
22 | 4, 21, 15, 5 | omndmul2 30713 | . . . 4 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ (𝑃 − 𝑁) ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
23 | 1, 14, 13, 20, 22 | syl121anc 1371 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
24 | eqid 2821 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
25 | 4, 21, 24 | omndadd 30707 | . . 3 ⊢ ((𝑀 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵 ∧ (𝑁 · 𝑋) ∈ 𝐵) ∧ 0 ≤ ((𝑃 − 𝑁) · 𝑋)) → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) ≤ (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
26 | 1, 7, 17, 19, 23, 25 | syl131anc 1379 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) ≤ (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
27 | 4, 24, 5 | mndlid 17931 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ (𝑁 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
28 | 3, 19, 27 | syl2anc 586 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
29 | 4, 15, 24 | mulgnn0dir 18257 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ ((𝑃 − 𝑁) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
30 | 3, 13, 8, 14, 29 | syl13anc 1368 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
31 | 9 | nn0cnd 11958 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
32 | 8 | nn0cnd 11958 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
33 | 31, 32 | npcand 11001 | . . . 4 ⊢ (𝜑 → ((𝑃 − 𝑁) + 𝑁) = 𝑃) |
34 | 33 | oveq1d 7171 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (𝑃 · 𝑋)) |
35 | 30, 34 | eqtr3d 2858 | . 2 ⊢ (𝜑 → (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋)) = (𝑃 · 𝑋)) |
36 | 26, 28, 35 | 3brtr3d 5097 | 1 ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 + caddc 10540 ≤ cle 10676 − cmin 10870 ℕ0cn0 11898 Basecbs 16483 +gcplusg 16565 lecple 16572 0gc0g 16713 Mndcmnd 17911 .gcmg 18224 oMndcomnd 30698 |
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-toset 17644 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mulg 18225 df-omnd 30700 |
This theorem is referenced by: (None) |
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