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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 10621 | . 2 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) | |
2 | addclnq 10583 | . 2 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
3 | dmplp 10650 | . 2 ⊢ dom +P = (P × P) | |
4 | addclpr 10656 | . 2 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P) | |
5 | addassnq 10596 | . 2 ⊢ ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)) | |
6 | 1, 2, 3, 4, 5 | genpass 10647 | 1 ⊢ ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 (class class class)co 7231 +Q cplq 10493 +P cpp 10499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-oadd 8226 df-omul 8227 df-er 8411 df-ni 10510 df-pli 10511 df-mi 10512 df-lti 10513 df-plpq 10546 df-mpq 10547 df-ltpq 10548 df-enq 10549 df-nq 10550 df-erq 10551 df-plq 10552 df-mq 10553 df-1nq 10554 df-rq 10555 df-ltnq 10556 df-np 10619 df-plp 10621 |
This theorem is referenced by: ltaprlem 10682 enrer 10701 addcmpblnr 10707 mulcmpblnrlem 10708 ltsrpr 10715 addasssr 10726 mulasssr 10728 distrsr 10729 m1p1sr 10730 m1m1sr 10731 ltsosr 10732 0idsr 10735 1idsr 10736 ltasr 10738 recexsrlem 10741 mulgt0sr 10743 map2psrpr 10748 |
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