Theorem omndadd2d 20349

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 20346	. . 3 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
3		tospos 18597	. . 3 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
4	1, 2, 3	3syl 19	. 2 ⊢ (𝜑 → 𝑀 ∈ Poset)
5		omndmnd 20345	. . . . 5 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
6	1, 5	syl 18	. . . 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 18936	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
12	6, 7, 8, 11	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
13		omndadd2d.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
14	9, 10	mndcl 18936	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
15	6, 13, 8, 14	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑍 + 𝑌) ∈ 𝐵)
16		omndadd2d.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
17	9, 10	mndcl 18936	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
18	6, 13, 16, 17	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
19	12, 15, 18	3jca 1146	. 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 20347	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑍) → (𝑋 + 𝑌) ≤ (𝑍 + 𝑌))
23	1, 7, 13, 8, 20, 22	syl131anc 1410	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑌))
24		omndadd2d.2	. . . 4 ⊢ (𝜑 → 𝑌 ≤ 𝑊)
25	9, 21, 10	omndadd 20347	. . . 4 ⊢ ((𝑀 ∈ oMnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑌 ≤ 𝑊) → (𝑌 + 𝑍) ≤ (𝑊 + 𝑍))
26	1, 8, 16, 13, 24, 25	syl131anc 1410	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ≤ (𝑊 + 𝑍))
27		omndadd2d.c	. . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd)
28	9, 10	cmncom 20017	. . . 4 ⊢ ((𝑀 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
29	27, 8, 13, 28	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
30	9, 10	cmncom 20017	. . . 4 ⊢ ((𝑀 ∈ CMnd ∧ 𝑊 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑊 + 𝑍) = (𝑍 + 𝑊))
31	27, 16, 13, 30	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑊 + 𝑍) = (𝑍 + 𝑊))
32	26, 29, 31	3brtr3d 5135	. 2 ⊢ (𝜑 → (𝑍 + 𝑌) ≤ (𝑍 + 𝑊))
33	9, 21	postr 18499	. . 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 412	. 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 852	1 ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  +_gcplusg 17433  lecple 17440  Posetcpo 18486  Tosetctos 18593  Mndcmnd 18928  CMndccmn 19999  oMndcomnd 20338

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-poset 18492  df-toset 18594  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-cmn 20001  df-omnd 20340

This theorem is used by:  omndmul  20354  gsumle  20364