Theorem omndadd 30046

Description: In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

Ref	Expression
omndadd.0	⊢ 𝐵 = (Base‘𝑀)
omndadd.1	⊢ ≤ = (le‘𝑀)
omndadd.2	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
omndadd	⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))

Proof of Theorem omndadd

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omndadd.0	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		omndadd.2	. . . . 5 ⊢ + = (+g‘𝑀)
3		omndadd.1	. . . . 5 ⊢ ≤ = (le‘𝑀)
4	1, 2, 3	isomnd 30041	. . . 4 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
5	4	simp3bi 1141	. . 3 ⊢ (𝑀 ∈ oMnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))
6		breq1 4789	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 ≤ 𝑏 ↔ 𝑋 ≤ 𝑏))
7		oveq1 6800	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 + 𝑐) = (𝑋 + 𝑐))
8	7	breq1d 4796	. . . . 5 ⊢ (𝑎 = 𝑋 → ((𝑎 + 𝑐) ≤ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≤ (𝑏 + 𝑐)))
9	6, 8	imbi12d 333	. . . 4 ⊢ (𝑎 = 𝑋 → ((𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) ↔ (𝑋 ≤ 𝑏 → (𝑋 + 𝑐) ≤ (𝑏 + 𝑐))))
10		breq2 4790	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 ≤ 𝑏 ↔ 𝑋 ≤ 𝑌))
11		oveq1 6800	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑏 + 𝑐) = (𝑌 + 𝑐))
12	11	breq2d 4798	. . . . 5 ⊢ (𝑏 = 𝑌 → ((𝑋 + 𝑐) ≤ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≤ (𝑌 + 𝑐)))
13	10, 12	imbi12d 333	. . . 4 ⊢ (𝑏 = 𝑌 → ((𝑋 ≤ 𝑏 → (𝑋 + 𝑐) ≤ (𝑏 + 𝑐)) ↔ (𝑋 ≤ 𝑌 → (𝑋 + 𝑐) ≤ (𝑌 + 𝑐))))
14		oveq2 6801	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15		oveq2 6801	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑌 + 𝑐) = (𝑌 + 𝑍))
16	14, 15	breq12d 4799	. . . . 5 ⊢ (𝑐 = 𝑍 → ((𝑋 + 𝑐) ≤ (𝑌 + 𝑐) ↔ (𝑋 + 𝑍) ≤ (𝑌 + 𝑍)))
17	16	imbi2d 329	. . . 4 ⊢ (𝑐 = 𝑍 → ((𝑋 ≤ 𝑌 → (𝑋 + 𝑐) ≤ (𝑌 + 𝑐)) ↔ (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))))
18	9, 13, 17	rspc3v 3475	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) → (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))))
19	5, 18	mpan9 496	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍)))
20	19	3impia 1109	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))

Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  +_gcplusg 16149  lecple 16156  Tosetctos 17241  Mndcmnd 17502  oMndcomnd 30037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-omnd 30039

This theorem is referenced by:  omndaddr  30047  omndadd2d  30048  omndadd2rd  30049  submomnd  30050  omndmul2  30052  omndmul3  30053  ogrpinvOLD  30055  ogrpinv0le  30056  ogrpsub  30057  ogrpaddlt  30058  orngsqr  30144  ornglmulle  30145  orngrmulle  30146