Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 30707 | . . . . 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 17928 | . . . 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 11950 | . . . . . 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 18246 | . . . 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 18246 | . . . 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 30715 | . . . 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 2823 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
25 | 4, 21, 24 | omndadd 30709 | . . 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 17933 | . . 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 18259 | . . . 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 11960 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
32 | 8 | nn0cnd 11960 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
33 | 31, 32 | npcand 11003 | . . . 4 ⊢ (𝜑 → ((𝑃 − 𝑁) + 𝑁) = 𝑃) |
34 | 33 | oveq1d 7173 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (𝑃 · 𝑋)) |
35 | 30, 34 | eqtr3d 2860 | . 2 ⊢ (𝜑 → (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋)) = (𝑃 · 𝑋)) |
36 | 26, 28, 35 | 3brtr3d 5099 | 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 5068 ‘cfv 6357 (class class class)co 7158 + caddc 10542 ≤ cle 10678 − cmin 10872 ℕ0cn0 11900 Basecbs 16485 +gcplusg 16567 lecple 16574 0gc0g 16715 Mndcmnd 17913 .gcmg 18226 oMndcomnd 30700 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-seq 13373 df-0g 16717 df-proset 17540 df-poset 17558 df-toset 17646 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mulg 18227 df-omnd 30702 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |