Theorem omndadd2d 31236

Description: In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

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)
omndadd2d.c	⊢ (𝜑 → 𝑀 ∈ CMnd)

Assertion

Ref	Expression
omndadd2d	⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))

Proof of Theorem omndadd2d

Step	Hyp	Ref	Expression
1		omndadd2d.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ oMnd)
2		omndtos 31233	. . 3 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
3		tospos 18053	. . 3 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
4	1, 2, 3	3syl 18	. 2 ⊢ (𝜑 → 𝑀 ∈ Poset)
5		omndmnd 31232	. . . . 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 18308	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
12	6, 7, 8, 11	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
13		omndadd2d.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
14	9, 10	mndcl 18308	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
15	6, 13, 8, 14	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑍 + 𝑌) ∈ 𝐵)
16		omndadd2d.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
17	9, 10	mndcl 18308	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
18	6, 13, 16, 17	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
19	12, 15, 18	3jca 1126	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵))
20		omndadd2d.1	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑍)
21		omndadd.1	. . . 4 ⊢ ≤ = (le‘𝑀)
22	9, 21, 10	omndadd 31234	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑍) → (𝑋 + 𝑌) ≤ (𝑍 + 𝑌))
23	1, 7, 13, 8, 20, 22	syl131anc 1381	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑌))
24		omndadd2d.2	. . . 4 ⊢ (𝜑 → 𝑌 ≤ 𝑊)
25	9, 21, 10	omndadd 31234	. . . 4 ⊢ ((𝑀 ∈ oMnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑌 ≤ 𝑊) → (𝑌 + 𝑍) ≤ (𝑊 + 𝑍))
26	1, 8, 16, 13, 24, 25	syl131anc 1381	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ≤ (𝑊 + 𝑍))
27		omndadd2d.c	. . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd)
28	9, 10	cmncom 19318	. . . 4 ⊢ ((𝑀 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
29	27, 8, 13, 28	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
30	9, 10	cmncom 19318	. . . 4 ⊢ ((𝑀 ∈ CMnd ∧ 𝑊 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑊 + 𝑍) = (𝑍 + 𝑊))
31	27, 16, 13, 30	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑊 + 𝑍) = (𝑍 + 𝑊))
32	26, 29, 31	3brtr3d 5101	. 2 ⊢ (𝜑 → (𝑍 + 𝑌) ≤ (𝑍 + 𝑊))
33	9, 21	postr 17953	. . 3 ⊢ ((𝑀 ∈ Poset ∧ ((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → (((𝑋 + 𝑌) ≤ (𝑍 + 𝑌) ∧ (𝑍 + 𝑌) ≤ (𝑍 + 𝑊)) → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊)))
34	33	imp 406	. 2 ⊢ (((𝑀 ∈ Poset ∧ ((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) ∧ ((𝑋 + 𝑌) ≤ (𝑍 + 𝑌) ∧ (𝑍 + 𝑌) ≤ (𝑍 + 𝑊))) → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))
35	4, 19, 23, 32, 34	syl22anc 835	1 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  lecple 16895  Posetcpo 17940  Tosetctos 18049  Mndcmnd 18300  CMndccmn 19301  oMndcomnd 31225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-poset 17946  df-toset 18050  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-cmn 19303  df-omnd 31227

This theorem is referenced by:  omndmul  31242  gsumle  31252