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Mirrors > Home > ILE Home > Th. List > addcomprg | Unicode version |
Description: Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Ref | Expression |
---|---|
addcomprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7251 | . . . . . . . . 9 | |
2 | elprnql 7257 | . . . . . . . . 9 | |
3 | 1, 2 | sylan 281 | . . . . . . . 8 |
4 | prop 7251 | . . . . . . . . . . . . 13 | |
5 | elprnql 7257 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | sylan 281 | . . . . . . . . . . . 12 |
7 | addcomnqg 7157 | . . . . . . . . . . . . 13 | |
8 | 7 | eqeq2d 2129 | . . . . . . . . . . . 12 |
9 | 6, 8 | sylan2 284 | . . . . . . . . . . 11 |
10 | 9 | anassrs 397 | . . . . . . . . . 10 |
11 | 10 | rexbidva 2411 | . . . . . . . . 9 |
12 | 11 | ancoms 266 | . . . . . . . 8 |
13 | 3, 12 | sylan2 284 | . . . . . . 7 |
14 | 13 | anassrs 397 | . . . . . 6 |
15 | 14 | rexbidva 2411 | . . . . 5 |
16 | rexcom 2572 | . . . . 5 | |
17 | 15, 16 | syl6bb 195 | . . . 4 |
18 | 17 | rabbidv 2649 | . . 3 |
19 | elprnqu 7258 | . . . . . . . . 9 | |
20 | 1, 19 | sylan 281 | . . . . . . . 8 |
21 | elprnqu 7258 | . . . . . . . . . . . . 13 | |
22 | 4, 21 | sylan 281 | . . . . . . . . . . . 12 |
23 | 22, 8 | sylan2 284 | . . . . . . . . . . 11 |
24 | 23 | anassrs 397 | . . . . . . . . . 10 |
25 | 24 | rexbidva 2411 | . . . . . . . . 9 |
26 | 25 | ancoms 266 | . . . . . . . 8 |
27 | 20, 26 | sylan2 284 | . . . . . . 7 |
28 | 27 | anassrs 397 | . . . . . 6 |
29 | 28 | rexbidva 2411 | . . . . 5 |
30 | rexcom 2572 | . . . . 5 | |
31 | 29, 30 | syl6bb 195 | . . . 4 |
32 | 31 | rabbidv 2649 | . . 3 |
33 | 18, 32 | opeq12d 3683 | . 2 |
34 | plpvlu 7314 | . . 3 | |
35 | 34 | ancoms 266 | . 2 |
36 | plpvlu 7314 | . 2 | |
37 | 33, 35, 36 | 3eqtr4rd 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wrex 2394 crab 2397 cop 3500 cfv 5093 (class class class)co 5742 c1st 6004 c2nd 6005 cnq 7056 cplq 7058 cnp 7067 cpp 7069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-plpq 7120 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-inp 7242 df-iplp 7244 |
This theorem is referenced by: prplnqu 7396 addextpr 7397 caucvgprlemcanl 7420 caucvgprprlemnkltj 7465 caucvgprprlemnbj 7469 caucvgprprlemmu 7471 caucvgprprlemloc 7479 caucvgprprlemexbt 7482 caucvgprprlemexb 7483 caucvgprprlemaddq 7484 enrer 7511 addcmpblnr 7515 mulcmpblnrlemg 7516 ltsrprg 7523 addcomsrg 7531 mulcomsrg 7533 mulasssrg 7534 distrsrg 7535 lttrsr 7538 ltposr 7539 ltsosr 7540 0lt1sr 7541 0idsr 7543 1idsr 7544 ltasrg 7546 recexgt0sr 7549 mulgt0sr 7554 aptisr 7555 mulextsr1lem 7556 archsr 7558 srpospr 7559 prsrpos 7561 prsradd 7562 prsrlt 7563 ltpsrprg 7579 map2psrprg 7581 pitonnlem1p1 7622 pitoregt0 7625 recidpirqlemcalc 7633 |
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