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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 10398 | . 2 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) | |
2 | addclnq 10360 | . 2 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
3 | dmplp 10427 | . 2 ⊢ dom +P = (P × P) | |
4 | addclpr 10433 | . 2 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P) | |
5 | addassnq 10373 | . 2 ⊢ ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)) | |
6 | 1, 2, 3, 4, 5 | genpass 10424 | 1 ⊢ ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 (class class class)co 7149 +Q cplq 10270 +P cpp 10276 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-omul 8100 df-er 8282 df-ni 10287 df-pli 10288 df-mi 10289 df-lti 10290 df-plpq 10323 df-mpq 10324 df-ltpq 10325 df-enq 10326 df-nq 10327 df-erq 10328 df-plq 10329 df-mq 10330 df-1nq 10331 df-rq 10332 df-ltnq 10333 df-np 10396 df-plp 10398 |
This theorem is referenced by: ltaprlem 10459 enrer 10478 addcmpblnr 10484 mulcmpblnrlem 10485 ltsrpr 10492 addasssr 10503 mulasssr 10505 distrsr 10506 m1p1sr 10507 m1m1sr 10508 ltsosr 10509 0idsr 10512 1idsr 10513 ltasr 10515 recexsrlem 10518 mulgt0sr 10520 map2psrpr 10525 |
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