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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omndaddr | Structured version Visualization version GIF version |
Description: In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
Ref | Expression |
---|---|
omndadd.0 | ⊢ 𝐵 = (Base‘𝑀) |
omndadd.1 | ⊢ ≤ = (le‘𝑀) |
omndadd.2 | ⊢ + = (+g‘𝑀) |
Ref | Expression |
---|---|
omndaddr | ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑍 + 𝑋) ≤ (𝑍 + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ (oppg‘𝑀) = (oppg‘𝑀) | |
2 | omndadd.0 | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
3 | 1, 2 | oppgbas 19387 | . . 3 ⊢ 𝐵 = (Base‘(oppg‘𝑀)) |
4 | omndadd.1 | . . . 4 ⊢ ≤ = (le‘𝑀) | |
5 | 1, 4 | oppgle 32925 | . . 3 ⊢ ≤ = (le‘(oppg‘𝑀)) |
6 | eqid 2734 | . . 3 ⊢ (+g‘(oppg‘𝑀)) = (+g‘(oppg‘𝑀)) | |
7 | 3, 5, 6 | omndadd 33048 | . 2 ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋(+g‘(oppg‘𝑀))𝑍) ≤ (𝑌(+g‘(oppg‘𝑀))𝑍)) |
8 | omndadd.2 | . . 3 ⊢ + = (+g‘𝑀) | |
9 | 8, 1, 6 | oppgplus 19384 | . 2 ⊢ (𝑋(+g‘(oppg‘𝑀))𝑍) = (𝑍 + 𝑋) |
10 | 8, 1, 6 | oppgplus 19384 | . 2 ⊢ (𝑌(+g‘(oppg‘𝑀))𝑍) = (𝑍 + 𝑌) |
11 | 7, 9, 10 | 3brtr3g 5202 | 1 ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑍 + 𝑋) ≤ (𝑍 + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 class class class wbr 5169 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 +gcplusg 17306 lecple 17313 oppgcoppg 19380 oMndcomnd 33039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-dec 12755 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-ple 17326 df-oppg 19381 df-omnd 33041 |
This theorem is referenced by: omndadd2rd 33051 |
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