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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 30755 | . . . . 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 17918 | . . . 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 11935 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) → (𝑁 ≤ 𝑃 ↔ (𝑃 − 𝑁) ∈ ℕ0)) | |
12 | 11 | biimpa 480 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) ∧ 𝑁 ≤ 𝑃) → (𝑃 − 𝑁) ∈ ℕ0) |
13 | 8, 9, 10, 12 | syl21anc 836 | . . . 4 ⊢ (𝜑 → (𝑃 − 𝑁) ∈ ℕ0) |
14 | omndmul3.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | omndmul3.m | . . . . 5 ⊢ · = (.g‘𝑀) | |
16 | 4, 15 | mulgnn0cl 18236 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ (𝑃 − 𝑁) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵) |
17 | 3, 13, 14, 16 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵) |
18 | 4, 15 | mulgnn0cl 18236 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
19 | 3, 8, 14, 18 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) |
20 | omndmul3.5 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝑋) | |
21 | omndmul.1 | . . . . 5 ⊢ ≤ = (le‘𝑀) | |
22 | 4, 21, 15, 5 | omndmul2 30763 | . . . 4 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ (𝑃 − 𝑁) ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
23 | 1, 14, 13, 20, 22 | syl121anc 1372 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
24 | eqid 2798 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
25 | 4, 21, 24 | omndadd 30757 | . . 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 1380 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) ≤ (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
27 | 4, 24, 5 | mndlid 17923 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ (𝑁 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
28 | 3, 19, 27 | syl2anc 587 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
29 | 4, 15, 24 | mulgnn0dir 18249 | . . . 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 1369 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
31 | 9 | nn0cnd 11945 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
32 | 8 | nn0cnd 11945 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
33 | 31, 32 | npcand 10990 | . . . 4 ⊢ (𝜑 → ((𝑃 − 𝑁) + 𝑁) = 𝑃) |
34 | 33 | oveq1d 7150 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (𝑃 · 𝑋)) |
35 | 30, 34 | eqtr3d 2835 | . 2 ⊢ (𝜑 → (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋)) = (𝑃 · 𝑋)) |
36 | 26, 28, 35 | 3brtr3d 5061 | 1 ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 + caddc 10529 ≤ cle 10665 − cmin 10859 ℕ0cn0 11885 Basecbs 16475 +gcplusg 16557 lecple 16564 0gc0g 16705 Mndcmnd 17903 .gcmg 18216 oMndcomnd 30748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-0g 16707 df-proset 17530 df-poset 17548 df-toset 17636 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mulg 18217 df-omnd 30750 |
This theorem is referenced by: (None) |
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