![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > addresr | Structured version Visualization version GIF version |
Description: Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addresr | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 + 〈𝐵, 0R〉) = 〈(𝐴 +R 𝐵), 0R〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 11123 | . . 3 ⊢ 0R ∈ R | |
2 | addcnsr 11178 | . . . 4 ⊢ (((𝐴 ∈ R ∧ 0R ∈ R) ∧ (𝐵 ∈ R ∧ 0R ∈ R)) → (〈𝐴, 0R〉 + 〈𝐵, 0R〉) = 〈(𝐴 +R 𝐵), (0R +R 0R)〉) | |
3 | 2 | an4s 658 | . . 3 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (0R ∈ R ∧ 0R ∈ R)) → (〈𝐴, 0R〉 + 〈𝐵, 0R〉) = 〈(𝐴 +R 𝐵), (0R +R 0R)〉) |
4 | 1, 1, 3 | mpanr12 703 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 + 〈𝐵, 0R〉) = 〈(𝐴 +R 𝐵), (0R +R 0R)〉) |
5 | 0idsr 11140 | . . . 4 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . 3 ⊢ (0R +R 0R) = 0R |
7 | 6 | opeq2i 4883 | . 2 ⊢ 〈(𝐴 +R 𝐵), (0R +R 0R)〉 = 〈(𝐴 +R 𝐵), 0R〉 |
8 | 4, 7 | eqtrdi 2782 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 + 〈𝐵, 0R〉) = 〈(𝐴 +R 𝐵), 0R〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 〈cop 4639 (class class class)co 7424 Rcnr 10908 0Rc0r 10909 +R cplr 10912 + caddc 11161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-oadd 8500 df-omul 8501 df-er 8734 df-ec 8736 df-qs 8740 df-ni 10915 df-pli 10916 df-mi 10917 df-lti 10918 df-plpq 10951 df-mpq 10952 df-ltpq 10953 df-enq 10954 df-nq 10955 df-erq 10956 df-plq 10957 df-mq 10958 df-1nq 10959 df-rq 10960 df-ltnq 10961 df-np 11024 df-1p 11025 df-plp 11026 df-ltp 11028 df-enr 11098 df-nr 11099 df-plr 11100 df-0r 11103 df-c 11164 df-add 11169 |
This theorem is referenced by: axaddrcl 11195 axi2m1 11202 axrnegex 11205 axpre-ltadd 11210 |
Copyright terms: Public domain | W3C validator |