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Mirrors > Home > ILE Home > Th. List > addassprg | GIF version |
Description: Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Ref | Expression |
---|---|
addassprg | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iplp 7390 | . 2 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ 〈{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦 +Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦 +Q 𝑧))}〉) | |
2 | addclnq 7297 | . 2 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
3 | dmplp 7462 | . 2 ⊢ dom +P = (P × P) | |
4 | addclpr 7459 | . 2 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P) | |
5 | addassnqg 7304 | . 2 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ))) | |
6 | 1, 2, 3, 4, 5 | genpassg 7448 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 (class class class)co 5826 +Q cplq 7204 Pcnp 7213 +P cpp 7215 |
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 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
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 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-eprel 4251 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-1o 6365 df-2o 6366 df-oadd 6369 df-omul 6370 df-er 6482 df-ec 6484 df-qs 6488 df-ni 7226 df-pli 7227 df-mi 7228 df-lti 7229 df-plpq 7266 df-mpq 7267 df-enq 7269 df-nqqs 7270 df-plqqs 7271 df-mqqs 7272 df-1nqqs 7273 df-rq 7274 df-ltnqqs 7275 df-enq0 7346 df-nq0 7347 df-0nq0 7348 df-plq0 7349 df-mq0 7350 df-inp 7388 df-iplp 7390 |
This theorem is referenced by: ltaprlem 7540 ltaprg 7541 caucvgprlemcanl 7566 caucvgprprlemexb 7629 caucvgprprlemaddq 7630 enrer 7657 addcmpblnr 7661 mulcmpblnrlemg 7662 ltsrprg 7669 addasssrg 7678 mulasssrg 7680 distrsrg 7681 m1p1sr 7682 m1m1sr 7683 lttrsr 7684 ltsosr 7686 0idsr 7689 1idsr 7690 ltasrg 7692 recexgt0sr 7695 mulgt0sr 7700 mulextsr1lem 7702 srpospr 7705 prsradd 7708 prsrlt 7709 map2psrprg 7727 pitonnlem1p1 7768 pitoregt0 7771 recidpirqlemcalc 7779 |
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