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Mirrors > Home > MPE Home > Th. List > axrnegex | Structured version Visualization version GIF version |
Description: Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 10679. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axrnegex | ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal2 10625 | . . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | |
2 | 1 | simplbi 501 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R) |
3 | m1r 10575 | . . . 4 ⊢ -1R ∈ R | |
4 | mulclsr 10577 | . . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 589 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R) |
6 | opelreal 10623 | . . 3 ⊢ (〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R) | |
7 | 5, 6 | sylibr 237 | . 2 ⊢ (𝐴 ∈ ℝ → 〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ) |
8 | 1 | simprbi 500 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
9 | 8 | oveq1d 7179 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) |
10 | addresr 10631 | . . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ ((1st ‘𝐴) ·R -1R) ∈ R) → (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉) | |
11 | 2, 5, 10 | syl2anc 587 | . . 3 ⊢ (𝐴 ∈ ℝ → (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉) |
12 | pn0sr 10594 | . . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R) | |
13 | 12 | opeq1d 4764 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ R → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 〈0R, 0R〉) |
14 | df-0 10615 | . . . . 5 ⊢ 0 = 〈0R, 0R〉 | |
15 | 13, 14 | eqtr4di 2791 | . . . 4 ⊢ ((1st ‘𝐴) ∈ R → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 0) |
16 | 2, 15 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 0) |
17 | 9, 11, 16 | 3eqtrd 2777 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) |
18 | oveq2 7172 | . . . 4 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → (𝐴 + 𝑥) = (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) | |
19 | 18 | eqeq1d 2740 | . . 3 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0)) |
20 | 19 | rspcev 3524 | . 2 ⊢ ((〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ∧ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
21 | 7, 17, 20 | syl2anc 587 | 1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 〈cop 4519 ‘cfv 6333 (class class class)co 7164 1st c1st 7705 Rcnr 10358 0Rc0r 10359 -1Rcm1r 10361 +R cplr 10362 ·R cmr 10363 ℝcr 10607 0cc0 10608 + caddc 10611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-oadd 8128 df-omul 8129 df-er 8313 df-ec 8315 df-qs 8319 df-ni 10365 df-pli 10366 df-mi 10367 df-lti 10368 df-plpq 10401 df-mpq 10402 df-ltpq 10403 df-enq 10404 df-nq 10405 df-erq 10406 df-plq 10407 df-mq 10408 df-1nq 10409 df-rq 10410 df-ltnq 10411 df-np 10474 df-1p 10475 df-plp 10476 df-mp 10477 df-ltp 10478 df-enr 10548 df-nr 10549 df-plr 10550 df-mr 10551 df-0r 10553 df-1r 10554 df-m1r 10555 df-c 10614 df-0 10615 df-r 10618 df-add 10619 |
This theorem is referenced by: (None) |
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