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Mirrors > Home > ILE Home > Th. List > addasssrg | GIF version |
Description: Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Ref | Expression |
---|---|
addasssrg | ⊢ ((A ∈ R ∧ B ∈ R ∧ 𝐶 ∈ R) → ((A +R B) +R 𝐶) = (A +R (B +R 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 6655 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | addsrpr 6673 | . 2 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → ([〈x, y〉] ~R +R [〈z, w〉] ~R ) = [〈(x +P z), (y +P w)〉] ~R ) | |
3 | addsrpr 6673 | . 2 ⊢ (((z ∈ P ∧ w ∈ P) ∧ (v ∈ P ∧ u ∈ P)) → ([〈z, w〉] ~R +R [〈v, u〉] ~R ) = [〈(z +P v), (w +P u)〉] ~R ) | |
4 | addsrpr 6673 | . 2 ⊢ ((((x +P z) ∈ P ∧ (y +P w) ∈ P) ∧ (v ∈ P ∧ u ∈ P)) → ([〈(x +P z), (y +P w)〉] ~R +R [〈v, u〉] ~R ) = [〈((x +P z) +P v), ((y +P w) +P u)〉] ~R ) | |
5 | addsrpr 6673 | . 2 ⊢ (((x ∈ P ∧ y ∈ P) ∧ ((z +P v) ∈ P ∧ (w +P u) ∈ P)) → ([〈x, y〉] ~R +R [〈(z +P v), (w +P u)〉] ~R ) = [〈(x +P (z +P v)), (y +P (w +P u))〉] ~R ) | |
6 | addclpr 6520 | . . . 4 ⊢ ((x ∈ P ∧ z ∈ P) → (x +P z) ∈ P) | |
7 | addclpr 6520 | . . . 4 ⊢ ((y ∈ P ∧ w ∈ P) → (y +P w) ∈ P) | |
8 | 6, 7 | anim12i 321 | . . 3 ⊢ (((x ∈ P ∧ z ∈ P) ∧ (y ∈ P ∧ w ∈ P)) → ((x +P z) ∈ P ∧ (y +P w) ∈ P)) |
9 | 8 | an4s 522 | . 2 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → ((x +P z) ∈ P ∧ (y +P w) ∈ P)) |
10 | addclpr 6520 | . . . 4 ⊢ ((z ∈ P ∧ v ∈ P) → (z +P v) ∈ P) | |
11 | addclpr 6520 | . . . 4 ⊢ ((w ∈ P ∧ u ∈ P) → (w +P u) ∈ P) | |
12 | 10, 11 | anim12i 321 | . . 3 ⊢ (((z ∈ P ∧ v ∈ P) ∧ (w ∈ P ∧ u ∈ P)) → ((z +P v) ∈ P ∧ (w +P u) ∈ P)) |
13 | 12 | an4s 522 | . 2 ⊢ (((z ∈ P ∧ w ∈ P) ∧ (v ∈ P ∧ u ∈ P)) → ((z +P v) ∈ P ∧ (w +P u) ∈ P)) |
14 | addassprg 6555 | . . . . 5 ⊢ ((x ∈ P ∧ z ∈ P ∧ v ∈ P) → ((x +P z) +P v) = (x +P (z +P v))) | |
15 | 14 | 3adant1r 1127 | . . . 4 ⊢ (((x ∈ P ∧ y ∈ P) ∧ z ∈ P ∧ v ∈ P) → ((x +P z) +P v) = (x +P (z +P v))) |
16 | 15 | 3adant2r 1129 | . . 3 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P) ∧ v ∈ P) → ((x +P z) +P v) = (x +P (z +P v))) |
17 | 16 | 3adant3r 1131 | . 2 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P) ∧ (v ∈ P ∧ u ∈ P)) → ((x +P z) +P v) = (x +P (z +P v))) |
18 | addassprg 6555 | . . . . 5 ⊢ ((y ∈ P ∧ w ∈ P ∧ u ∈ P) → ((y +P w) +P u) = (y +P (w +P u))) | |
19 | 18 | 3adant1l 1126 | . . . 4 ⊢ (((x ∈ P ∧ y ∈ P) ∧ w ∈ P ∧ u ∈ P) → ((y +P w) +P u) = (y +P (w +P u))) |
20 | 19 | 3adant2l 1128 | . . 3 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P) ∧ u ∈ P) → ((y +P w) +P u) = (y +P (w +P u))) |
21 | 20 | 3adant3l 1130 | . 2 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P) ∧ (v ∈ P ∧ u ∈ P)) → ((y +P w) +P u) = (y +P (w +P u))) |
22 | 1, 2, 3, 4, 5, 9, 13, 17, 21 | ecoviass 6152 | 1 ⊢ ((A ∈ R ∧ B ∈ R ∧ 𝐶 ∈ R) → ((A +R B) +R 𝐶) = (A +R (B +R 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 (class class class)co 5455 Pcnp 6275 +P cpp 6277 ~R cer 6280 Rcnr 6281 +R cplr 6285 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-iplp 6451 df-enr 6654 df-nr 6655 df-plr 6656 |
This theorem is referenced by: axaddass 6756 axmulass 6757 axdistr 6758 |
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