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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 10980 | . 2 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) | |
2 | addclnq 10942 | . 2 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
3 | dmplp 11009 | . 2 ⊢ dom +P = (P × P) | |
4 | addclpr 11015 | . 2 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P) | |
5 | addassnq 10955 | . 2 ⊢ ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)) | |
6 | 1, 2, 3, 4, 5 | genpass 11006 | 1 ⊢ ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 (class class class)co 7411 +Q cplq 10852 +P cpp 10858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-omul 8473 df-er 8705 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-mpq 10906 df-ltpq 10907 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-mq 10912 df-1nq 10913 df-rq 10914 df-ltnq 10915 df-np 10978 df-plp 10980 |
This theorem is referenced by: ltaprlem 11041 enrer 11060 addcmpblnr 11066 mulcmpblnrlem 11067 ltsrpr 11074 addasssr 11085 mulasssr 11087 distrsr 11088 m1p1sr 11089 m1m1sr 11090 ltsosr 11091 0idsr 11094 1idsr 11095 ltasr 11097 recexsrlem 11100 mulgt0sr 11102 map2psrpr 11107 |
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