Mathbox for Thierry Arnoux |
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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 30633 | . . 3 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) | |
3 | tospos 30572 | . . 3 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑀 ∈ Poset) |
5 | omndmnd 30632 | . . . . 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 17907 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
12 | 6, 7, 8, 11 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
13 | omndadd2d.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) | |
14 | 9, 10 | mndcl 17907 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 + 𝑊) ∈ 𝐵) |
15 | 6, 7, 13, 14 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑊) ∈ 𝐵) |
16 | omndadd2d.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
17 | 9, 10 | mndcl 17907 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵) |
18 | 6, 16, 13, 17 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵) |
19 | 12, 15, 18 | 3jca 1120 | . 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 30635 | . . 3 ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑌 ≤ 𝑊) → (𝑋 + 𝑌) ≤ (𝑋 + 𝑊)) |
24 | 20, 8, 13, 7, 21, 23 | syl131anc 1375 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑋 + 𝑊)) |
25 | omndadd2d.1 | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑍) | |
26 | 9, 22, 10 | omndadd 30634 | . . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝑋 ≤ 𝑍) → (𝑋 + 𝑊) ≤ (𝑍 + 𝑊)) |
27 | 1, 7, 16, 13, 25, 26 | syl131anc 1375 | . 2 ⊢ (𝜑 → (𝑋 + 𝑊) ≤ (𝑍 + 𝑊)) |
28 | 9, 22 | postr 17551 | . . 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 407 | . 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 834 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 lecple 16560 Posetcpo 17538 Tosetctos 17631 Mndcmnd 17899 oppgcoppg 18411 oMndcomnd 30625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-ple 16573 df-poset 17544 df-toset 17632 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-oppg 18412 df-omnd 30627 |
This theorem is referenced by: archiabllem2c 30751 |
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