Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7395 | . . . . . . . . 9 | |
2 | elprnql 7401 | . . . . . . . . 9 | |
3 | 1, 2 | sylan 281 | . . . . . . . 8 |
4 | prop 7395 | . . . . . . . . . . . . 13 | |
5 | elprnql 7401 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | sylan 281 | . . . . . . . . . . . 12 |
7 | addcomnqg 7301 | . . . . . . . . . . . . 13 | |
8 | 7 | eqeq2d 2169 | . . . . . . . . . . . 12 |
9 | 6, 8 | sylan2 284 | . . . . . . . . . . 11 |
10 | 9 | anassrs 398 | . . . . . . . . . 10 |
11 | 10 | rexbidva 2454 | . . . . . . . . 9 |
12 | 11 | ancoms 266 | . . . . . . . 8 |
13 | 3, 12 | sylan2 284 | . . . . . . 7 |
14 | 13 | anassrs 398 | . . . . . 6 |
15 | 14 | rexbidva 2454 | . . . . 5 |
16 | rexcom 2621 | . . . . 5 | |
17 | 15, 16 | bitrdi 195 | . . . 4 |
18 | 17 | rabbidv 2701 | . . 3 |
19 | elprnqu 7402 | . . . . . . . . 9 | |
20 | 1, 19 | sylan 281 | . . . . . . . 8 |
21 | elprnqu 7402 | . . . . . . . . . . . . 13 | |
22 | 4, 21 | sylan 281 | . . . . . . . . . . . 12 |
23 | 22, 8 | sylan2 284 | . . . . . . . . . . 11 |
24 | 23 | anassrs 398 | . . . . . . . . . 10 |
25 | 24 | rexbidva 2454 | . . . . . . . . 9 |
26 | 25 | ancoms 266 | . . . . . . . 8 |
27 | 20, 26 | sylan2 284 | . . . . . . 7 |
28 | 27 | anassrs 398 | . . . . . 6 |
29 | 28 | rexbidva 2454 | . . . . 5 |
30 | rexcom 2621 | . . . . 5 | |
31 | 29, 30 | bitrdi 195 | . . . 4 |
32 | 31 | rabbidv 2701 | . . 3 |
33 | 18, 32 | opeq12d 3749 | . 2 |
34 | plpvlu 7458 | . . 3 | |
35 | 34 | ancoms 266 | . 2 |
36 | plpvlu 7458 | . 2 | |
37 | 33, 35, 36 | 3eqtr4rd 2201 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wrex 2436 crab 2439 cop 3563 cfv 5170 (class class class)co 5824 c1st 6086 c2nd 6087 cnq 7200 cplq 7202 cnp 7211 cpp 7213 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-plpq 7264 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-inp 7386 df-iplp 7388 |
This theorem is referenced by: prplnqu 7540 addextpr 7541 caucvgprlemcanl 7564 caucvgprprlemnkltj 7609 caucvgprprlemnbj 7613 caucvgprprlemmu 7615 caucvgprprlemloc 7623 caucvgprprlemexbt 7626 caucvgprprlemexb 7627 caucvgprprlemaddq 7628 enrer 7655 addcmpblnr 7659 mulcmpblnrlemg 7660 ltsrprg 7667 addcomsrg 7675 mulcomsrg 7677 mulasssrg 7678 distrsrg 7679 lttrsr 7682 ltposr 7683 ltsosr 7684 0lt1sr 7685 0idsr 7687 1idsr 7688 ltasrg 7690 recexgt0sr 7693 mulgt0sr 7698 aptisr 7699 mulextsr1lem 7700 archsr 7702 srpospr 7703 prsrpos 7705 prsradd 7706 prsrlt 7707 ltpsrprg 7723 map2psrprg 7725 pitonnlem1p1 7766 pitoregt0 7769 recidpirqlemcalc 7777 |
Copyright terms: Public domain | W3C validator |