![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > omndadd2rd | Structured version Visualization version GIF version |
Description: In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
omndadd.0 | ⊢ 𝐵 = (Base‘𝑀) |
omndadd.1 | ⊢ ≤ = (le‘𝑀) |
omndadd.2 | ⊢ + = (+g‘𝑀) |
omndadd2d.m | ⊢ (𝜑 → 𝑀 ∈ oMnd) |
omndadd2d.w | ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
omndadd2d.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
omndadd2d.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
omndadd2d.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
omndadd2d.1 | ⊢ (𝜑 → 𝑋 ≤ 𝑍) |
omndadd2d.2 | ⊢ (𝜑 → 𝑌 ≤ 𝑊) |
omndadd2rd.c | ⊢ (𝜑 → (oppg‘𝑀) ∈ oMnd) |
Ref | Expression |
---|---|
omndadd2rd | ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omndadd2d.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ oMnd) | |
2 | omndtos 30756 | . . 3 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) | |
3 | tospos 30671 | . . 3 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑀 ∈ Poset) |
5 | omndmnd 30755 | . . . . 5 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) | |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd) |
7 | omndadd2d.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | omndadd2d.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | omndadd.0 | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
10 | omndadd.2 | . . . . 5 ⊢ + = (+g‘𝑀) | |
11 | 9, 10 | mndcl 17911 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
12 | 6, 7, 8, 11 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
13 | omndadd2d.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) | |
14 | 9, 10 | mndcl 17911 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 + 𝑊) ∈ 𝐵) |
15 | 6, 7, 13, 14 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑊) ∈ 𝐵) |
16 | omndadd2d.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
17 | 9, 10 | mndcl 17911 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵) |
18 | 6, 16, 13, 17 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵) |
19 | 12, 15, 18 | 3jca 1125 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑋 + 𝑊) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) |
20 | omndadd2rd.c | . . 3 ⊢ (𝜑 → (oppg‘𝑀) ∈ oMnd) | |
21 | omndadd2d.2 | . . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊) | |
22 | omndadd.1 | . . . 4 ⊢ ≤ = (le‘𝑀) | |
23 | 9, 22, 10 | omndaddr 30758 | . . 3 ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑌 ≤ 𝑊) → (𝑋 + 𝑌) ≤ (𝑋 + 𝑊)) |
24 | 20, 8, 13, 7, 21, 23 | syl131anc 1380 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑋 + 𝑊)) |
25 | omndadd2d.1 | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑍) | |
26 | 9, 22, 10 | omndadd 30757 | . . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝑋 ≤ 𝑍) → (𝑋 + 𝑊) ≤ (𝑍 + 𝑊)) |
27 | 1, 7, 16, 13, 25, 26 | syl131anc 1380 | . 2 ⊢ (𝜑 → (𝑋 + 𝑊) ≤ (𝑍 + 𝑊)) |
28 | 9, 22 | postr 17555 | . . 3 ⊢ ((𝑀 ∈ Poset ∧ ((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑋 + 𝑊) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → (((𝑋 + 𝑌) ≤ (𝑋 + 𝑊) ∧ (𝑋 + 𝑊) ≤ (𝑍 + 𝑊)) → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))) |
29 | 28 | imp 410 | . 2 ⊢ (((𝑀 ∈ Poset ∧ ((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑋 + 𝑊) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) ∧ ((𝑋 + 𝑌) ≤ (𝑋 + 𝑊) ∧ (𝑋 + 𝑊) ≤ (𝑍 + 𝑊))) → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊)) |
30 | 4, 19, 24, 27, 29 | syl22anc 837 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 lecple 16564 Posetcpo 17542 Tosetctos 17635 Mndcmnd 17903 oppgcoppg 18465 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-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-tpos 7875 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-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-ple 16577 df-poset 17548 df-toset 17636 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-oppg 18466 df-omnd 30750 |
This theorem is referenced by: archiabllem2c 30874 |
Copyright terms: Public domain | W3C validator |