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 29035 | . . . . 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 17131 | . . . 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 11220 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) → (𝑁 ≤ 𝑃 ↔ (𝑃 − 𝑁) ∈ ℕ0)) | |
12 | 11 | biimpa 500 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) ∧ 𝑁 ≤ 𝑃) → (𝑃 − 𝑁) ∈ ℕ0) |
13 | 8, 9, 10, 12 | syl21anc 1317 | . . . 4 ⊢ (𝜑 → (𝑃 − 𝑁) ∈ ℕ0) |
14 | omndmul3.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | omndmul3.m | . . . . 5 ⊢ · = (.g‘𝑀) | |
16 | 4, 15 | mulgnn0cl 17381 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ (𝑃 − 𝑁) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵) |
17 | 3, 13, 14, 16 | syl3anc 1318 | . . 3 ⊢ (𝜑 → ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵) |
18 | 4, 15 | mulgnn0cl 17381 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
19 | 3, 8, 14, 18 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) |
20 | omndmul3.5 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝑋) | |
21 | omndmul.1 | . . . . 5 ⊢ ≤ = (le‘𝑀) | |
22 | 4, 21, 15, 5 | omndmul2 29043 | . . . 4 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ (𝑃 − 𝑁) ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
23 | 1, 14, 13, 20, 22 | syl121anc 1323 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
24 | eqid 2610 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
25 | 4, 21, 24 | omndadd 29037 | . . 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 1331 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) ≤ (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
27 | 4, 24, 5 | mndlid 17134 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ (𝑁 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
28 | 3, 19, 27 | syl2anc 691 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
29 | 4, 15, 24 | mulgnn0dir 17394 | . . . 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 1320 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
31 | 9 | nn0cnd 11230 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
32 | 8 | nn0cnd 11230 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
33 | 31, 32 | npcand 10275 | . . . 4 ⊢ (𝜑 → ((𝑃 − 𝑁) + 𝑁) = 𝑃) |
34 | 33 | oveq1d 6564 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (𝑃 · 𝑋)) |
35 | 30, 34 | eqtr3d 2646 | . 2 ⊢ (𝜑 → (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋)) = (𝑃 · 𝑋)) |
36 | 26, 28, 35 | 3brtr3d 4614 | 1 ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕ0cn0 11169 Basecbs 15695 +gcplusg 15768 lecple 15775 0gc0g 15923 Mndcmnd 17117 .gcmg 17363 oMndcomnd 29028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 df-0g 15925 df-preset 16751 df-poset 16769 df-toset 16857 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mulg 17364 df-omnd 29030 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |