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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 10595 be used later. Instead, use addass 10617. (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 10557 | . 2 ⊢ ℂ = ((R × R) / ◡ E ) | |
2 | addcnsrec 10558 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ([〈𝑥, 𝑦〉]◡ E + [〈𝑧, 𝑤〉]◡ E ) = [〈(𝑥 +R 𝑧), (𝑦 +R 𝑤)〉]◡ E ) | |
3 | addcnsrec 10558 | . 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ([〈𝑧, 𝑤〉]◡ E + [〈𝑣, 𝑢〉]◡ E ) = [〈(𝑧 +R 𝑣), (𝑤 +R 𝑢)〉]◡ E ) | |
4 | addcnsrec 10558 | . 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 10558 | . 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 10498 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) ∈ R) | |
7 | addclsr 10498 | . . . 4 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) ∈ R) | |
8 | 6, 7 | anim12i 614 | . . 3 ⊢ (((𝑥 ∈ R ∧ 𝑧 ∈ R) ∧ (𝑦 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R)) |
9 | 8 | an4s 658 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R)) |
10 | addclsr 10498 | . . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑣 ∈ R) → (𝑧 +R 𝑣) ∈ R) | |
11 | addclsr 10498 | . . . 4 ⊢ ((𝑤 ∈ R ∧ 𝑢 ∈ R) → (𝑤 +R 𝑢) ∈ R) | |
12 | 10, 11 | anim12i 614 | . . 3 ⊢ (((𝑧 ∈ R ∧ 𝑣 ∈ R) ∧ (𝑤 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R)) |
13 | 12 | an4s 658 | . 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R)) |
14 | addasssr 10503 | . 2 ⊢ ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣)) | |
15 | addasssr 10503 | . 2 ⊢ ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢)) | |
16 | 1, 2, 3, 4, 5, 9, 13, 14, 15 | ecovass 8397 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 E cep 5457 ◡ccnv 5547 (class class class)co 7149 Rcnr 10280 +R cplr 10284 ℂcc 10528 + caddc 10533 |
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-int 4870 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-ec 8284 df-qs 8288 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 df-ltp 10400 df-enr 10470 df-nr 10471 df-plr 10472 df-c 10536 df-add 10541 |
This theorem is referenced by: (None) |
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