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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 10794 be used later. Instead, use addass 10816. (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 10756 | . 2 ⊢ ℂ = ((R × R) / ◡ E ) | |
2 | addcnsrec 10757 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ([〈𝑥, 𝑦〉]◡ E + [〈𝑧, 𝑤〉]◡ E ) = [〈(𝑥 +R 𝑧), (𝑦 +R 𝑤)〉]◡ E ) | |
3 | addcnsrec 10757 | . 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ([〈𝑧, 𝑤〉]◡ E + [〈𝑣, 𝑢〉]◡ E ) = [〈(𝑧 +R 𝑣), (𝑤 +R 𝑢)〉]◡ E ) | |
4 | addcnsrec 10757 | . 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 10757 | . 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 10697 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) ∈ R) | |
7 | addclsr 10697 | . . . 4 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) ∈ R) | |
8 | 6, 7 | anim12i 616 | . . 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 10697 | . . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑣 ∈ R) → (𝑧 +R 𝑣) ∈ R) | |
11 | addclsr 10697 | . . . 4 ⊢ ((𝑤 ∈ R ∧ 𝑢 ∈ R) → (𝑤 +R 𝑢) ∈ R) | |
12 | 10, 11 | anim12i 616 | . . 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 10702 | . 2 ⊢ ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣)) | |
15 | addasssr 10702 | . 2 ⊢ ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢)) | |
16 | 1, 2, 3, 4, 5, 9, 13, 14, 15 | ecovass 8506 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 E cep 5459 ◡ccnv 5550 (class class class)co 7213 Rcnr 10479 +R cplr 10483 ℂcc 10727 + caddc 10732 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-omul 8207 df-er 8391 df-ec 8393 df-qs 8397 df-ni 10486 df-pli 10487 df-mi 10488 df-lti 10489 df-plpq 10522 df-mpq 10523 df-ltpq 10524 df-enq 10525 df-nq 10526 df-erq 10527 df-plq 10528 df-mq 10529 df-1nq 10530 df-rq 10531 df-ltnq 10532 df-np 10595 df-plp 10597 df-ltp 10599 df-enr 10669 df-nr 10670 df-plr 10671 df-c 10735 df-add 10740 |
This theorem is referenced by: (None) |
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