![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axaddass | Structured version Visualization version GIF version |
Description: Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 11217 be used later. Instead, use addass 11239. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axaddass | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnqs 11179 | . 2 ⊢ ℂ = ((R × R) / ◡ E ) | |
2 | addcnsrec 11180 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ([〈𝑥, 𝑦〉]◡ E + [〈𝑧, 𝑤〉]◡ E ) = [〈(𝑥 +R 𝑧), (𝑦 +R 𝑤)〉]◡ E ) | |
3 | addcnsrec 11180 | . 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ([〈𝑧, 𝑤〉]◡ E + [〈𝑣, 𝑢〉]◡ E ) = [〈(𝑧 +R 𝑣), (𝑤 +R 𝑢)〉]◡ E ) | |
4 | addcnsrec 11180 | . 2 ⊢ ((((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ([〈(𝑥 +R 𝑧), (𝑦 +R 𝑤)〉]◡ E + [〈𝑣, 𝑢〉]◡ E ) = [〈((𝑥 +R 𝑧) +R 𝑣), ((𝑦 +R 𝑤) +R 𝑢)〉]◡ E ) | |
5 | addcnsrec 11180 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R)) → ([〈𝑥, 𝑦〉]◡ E + [〈(𝑧 +R 𝑣), (𝑤 +R 𝑢)〉]◡ E ) = [〈(𝑥 +R (𝑧 +R 𝑣)), (𝑦 +R (𝑤 +R 𝑢))〉]◡ E ) | |
6 | addclsr 11120 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) ∈ R) | |
7 | addclsr 11120 | . . . 4 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) ∈ R) | |
8 | 6, 7 | anim12i 613 | . . 3 ⊢ (((𝑥 ∈ R ∧ 𝑧 ∈ R) ∧ (𝑦 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R)) |
9 | 8 | an4s 660 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R)) |
10 | addclsr 11120 | . . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑣 ∈ R) → (𝑧 +R 𝑣) ∈ R) | |
11 | addclsr 11120 | . . . 4 ⊢ ((𝑤 ∈ R ∧ 𝑢 ∈ R) → (𝑤 +R 𝑢) ∈ R) | |
12 | 10, 11 | anim12i 613 | . . 3 ⊢ (((𝑧 ∈ R ∧ 𝑣 ∈ R) ∧ (𝑤 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R)) |
13 | 12 | an4s 660 | . 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R)) |
14 | addasssr 11125 | . 2 ⊢ ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣)) | |
15 | addasssr 11125 | . 2 ⊢ ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢)) | |
16 | 1, 2, 3, 4, 5, 9, 13, 14, 15 | ecovass 8862 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 E cep 5587 ◡ccnv 5687 (class class class)co 7430 Rcnr 10902 +R cplr 10906 ℂcc 11150 + caddc 11155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-omul 8509 df-er 8743 df-ec 8745 df-qs 8749 df-ni 10909 df-pli 10910 df-mi 10911 df-lti 10912 df-plpq 10945 df-mpq 10946 df-ltpq 10947 df-enq 10948 df-nq 10949 df-erq 10950 df-plq 10951 df-mq 10952 df-1nq 10953 df-rq 10954 df-ltnq 10955 df-np 11018 df-plp 11020 df-ltp 11022 df-enr 11092 df-nr 11093 df-plr 11094 df-c 11158 df-add 11163 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |