![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > addasspr | Structured version Visualization version GIF version |
Description: Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addasspr | ⊢ ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-plp 10977 | . 2 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) | |
2 | addclnq 10939 | . 2 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
3 | dmplp 11006 | . 2 ⊢ dom +P = (P × P) | |
4 | addclpr 11012 | . 2 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P) | |
5 | addassnq 10952 | . 2 ⊢ ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)) | |
6 | 1, 2, 3, 4, 5 | genpass 11003 | 1 ⊢ ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7408 +Q cplq 10849 +P cpp 10855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-omul 8470 df-er 8702 df-ni 10866 df-pli 10867 df-mi 10868 df-lti 10869 df-plpq 10902 df-mpq 10903 df-ltpq 10904 df-enq 10905 df-nq 10906 df-erq 10907 df-plq 10908 df-mq 10909 df-1nq 10910 df-rq 10911 df-ltnq 10912 df-np 10975 df-plp 10977 |
This theorem is referenced by: ltaprlem 11038 enrer 11057 addcmpblnr 11063 mulcmpblnrlem 11064 ltsrpr 11071 addasssr 11082 mulasssr 11084 distrsr 11085 m1p1sr 11086 m1m1sr 11087 ltsosr 11088 0idsr 11091 1idsr 11092 ltasr 11094 recexsrlem 11097 mulgt0sr 11099 map2psrpr 11104 |
Copyright terms: Public domain | W3C validator |