Theorem omndadd2d 30759

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 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.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
14	9, 10	mndcl 17911	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
15	6, 13, 8, 14	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑍 + 𝑌) ∈ 𝐵)
16		omndadd2d.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
17	9, 10	mndcl 17911	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
18	6, 13, 16, 17	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
19	12, 15, 18	3jca 1125	. 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 30757	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑍) → (𝑋 + 𝑌) ≤ (𝑍 + 𝑌))
23	1, 7, 13, 8, 20, 22	syl131anc 1380	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑌))
24		omndadd2d.2	. . . 4 ⊢ (𝜑 → 𝑌 ≤ 𝑊)
25	9, 21, 10	omndadd 30757	. . . 4 ⊢ ((𝑀 ∈ oMnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑌 ≤ 𝑊) → (𝑌 + 𝑍) ≤ (𝑊 + 𝑍))
26	1, 8, 16, 13, 24, 25	syl131anc 1380	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ≤ (𝑊 + 𝑍))
27		omndadd2d.c	. . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd)
28	9, 10	cmncom 18915	. . . 4 ⊢ ((𝑀 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
29	27, 8, 13, 28	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
30	9, 10	cmncom 18915	. . . 4 ⊢ ((𝑀 ∈ CMnd ∧ 𝑊 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑊 + 𝑍) = (𝑍 + 𝑊))
31	27, 16, 13, 30	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑊 + 𝑍) = (𝑍 + 𝑊))
32	26, 29, 31	3brtr3d 5061	. 2 ⊢ (𝜑 → (𝑍 + 𝑌) ≤ (𝑍 + 𝑊))
33	9, 21	postr 17555	. . 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 410	. 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 837	1 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))

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  CMndccmn 18898  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-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-poset 17548  df-toset 17636  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-cmn 18900  df-omnd 30750

This theorem is referenced by:  omndmul  30765  gsumle  30775