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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 32660 | . . . . 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 18680 | . . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵) |
7 | 3, 6 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) |
8 | omndmul3.m | . . . 4 ⊢ · = (.g‘𝑀) | |
9 | omndmul3.1 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
10 | omndmul3.2 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ0) | |
11 | omndmul3.3 | . . . . 5 ⊢ (𝜑 → 𝑁 ≤ 𝑃) | |
12 | nn0sub 12529 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) → (𝑁 ≤ 𝑃 ↔ (𝑃 − 𝑁) ∈ ℕ0)) | |
13 | 12 | biimpa 476 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℕ0) ∧ 𝑁 ≤ 𝑃) → (𝑃 − 𝑁) ∈ ℕ0) |
14 | 9, 10, 11, 13 | syl21anc 835 | . . . 4 ⊢ (𝜑 → (𝑃 − 𝑁) ∈ ℕ0) |
15 | omndmul3.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
16 | 4, 8, 3, 14, 15 | mulgnn0cld 19018 | . . 3 ⊢ (𝜑 → ((𝑃 − 𝑁) · 𝑋) ∈ 𝐵) |
17 | 4, 8, 3, 9, 15 | mulgnn0cld 19018 | . . 3 ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) |
18 | omndmul3.5 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝑋) | |
19 | omndmul.1 | . . . . 5 ⊢ ≤ = (le‘𝑀) | |
20 | 4, 19, 8, 5 | omndmul2 32668 | . . . 4 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ (𝑃 − 𝑁) ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
21 | 1, 15, 14, 18, 20 | syl121anc 1374 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑃 − 𝑁) · 𝑋)) |
22 | eqid 2731 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
23 | 4, 19, 22 | omndadd 32662 | . . 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))) |
24 | 1, 7, 16, 17, 21, 23 | syl131anc 1382 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) ≤ (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
25 | 4, 22, 5 | mndlid 18685 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ (𝑁 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
26 | 3, 17, 25 | syl2anc 583 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑀)(𝑁 · 𝑋)) = (𝑁 · 𝑋)) |
27 | 4, 8, 22 | mulgnn0dir 19027 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ ((𝑃 − 𝑁) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
28 | 3, 14, 9, 15, 27 | syl13anc 1371 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋))) |
29 | 10 | nn0cnd 12541 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
30 | 9 | nn0cnd 12541 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
31 | 29, 30 | npcand 11582 | . . . 4 ⊢ (𝜑 → ((𝑃 − 𝑁) + 𝑁) = 𝑃) |
32 | 31 | oveq1d 7427 | . . 3 ⊢ (𝜑 → (((𝑃 − 𝑁) + 𝑁) · 𝑋) = (𝑃 · 𝑋)) |
33 | 28, 32 | eqtr3d 2773 | . 2 ⊢ (𝜑 → (((𝑃 − 𝑁) · 𝑋)(+g‘𝑀)(𝑁 · 𝑋)) = (𝑃 · 𝑋)) |
34 | 24, 26, 33 | 3brtr3d 5179 | 1 ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 + caddc 11119 ≤ cle 11256 − cmin 11451 ℕ0cn0 12479 Basecbs 17151 +gcplusg 17204 lecple 17211 0gc0g 17392 Mndcmnd 18665 .gcmg 18993 oMndcomnd 32653 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-seq 13974 df-0g 17394 df-proset 18258 df-poset 18276 df-toset 18380 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mulg 18994 df-omnd 32655 |
This theorem is referenced by: (None) |
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