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Mirrors > Home > ILE Home > Th. List > axrnegex | GIF version |
Description: Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 7883. (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 7792 | . . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | |
2 | 1 | simplbi 272 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R) |
3 | m1r 7714 | . . . 4 ⊢ -1R ∈ R | |
4 | mulclsr 7716 | . . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 411 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R) |
6 | opelreal 7789 | . . 3 ⊢ (〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R) | |
7 | 5, 6 | sylibr 133 | . 2 ⊢ (𝐴 ∈ ℝ → 〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ) |
8 | 1 | simprbi 273 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
9 | 8 | oveq1d 5868 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) |
10 | addresr 7799 | . . . 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 409 | . . 3 ⊢ (𝐴 ∈ ℝ → (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉) |
12 | pn0sr 7733 | . . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R) | |
13 | 12 | opeq1d 3771 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ R → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 〈0R, 0R〉) |
14 | df-0 7781 | . . . . 5 ⊢ 0 = 〈0R, 0R〉 | |
15 | 13, 14 | eqtr4di 2221 | . . . 4 ⊢ ((1st ‘𝐴) ∈ R → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 0) |
16 | 2, 15 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℝ → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 0) |
17 | 9, 11, 16 | 3eqtrd 2207 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) |
18 | oveq2 5861 | . . . 4 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → (𝐴 + 𝑥) = (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) | |
19 | 18 | eqeq1d 2179 | . . 3 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0)) |
20 | 19 | rspcev 2834 | . 2 ⊢ ((〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ∧ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
21 | 7, 17, 20 | syl2anc 409 | 1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ∃wrex 2449 〈cop 3586 ‘cfv 5198 (class class class)co 5853 1st c1st 6117 Rcnr 7259 0Rc0r 7260 -1Rcm1r 7262 +R cplr 7263 ·R cmr 7264 ℝcr 7773 0cc0 7774 + caddc 7777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-i1p 7429 df-iplp 7430 df-imp 7431 df-enr 7688 df-nr 7689 df-plr 7690 df-mr 7691 df-0r 7693 df-1r 7694 df-m1r 7695 df-c 7780 df-0 7781 df-r 7784 df-add 7785 |
This theorem is referenced by: (None) |
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