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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 11178. (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 11124 | . . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | |
2 | 1 | simplbi 497 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R) |
3 | m1r 11074 | . . . 4 ⊢ -1R ∈ R | |
4 | mulclsr 11076 | . . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 585 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R) |
6 | opelreal 11122 | . . 3 ⊢ (〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R) | |
7 | 5, 6 | sylibr 233 | . 2 ⊢ (𝐴 ∈ ℝ → 〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ) |
8 | 1 | simprbi 496 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
9 | 8 | oveq1d 7417 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) |
10 | addresr 11130 | . . . 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 583 | . . 3 ⊢ (𝐴 ∈ ℝ → (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉) |
12 | pn0sr 11093 | . . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R) | |
13 | 12 | opeq1d 4872 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ R → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 〈0R, 0R〉) |
14 | df-0 11114 | . . . . 5 ⊢ 0 = 〈0R, 0R〉 | |
15 | 13, 14 | eqtr4di 2782 | . . . 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 2768 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) |
18 | oveq2 7410 | . . . 4 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → (𝐴 + 𝑥) = (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) | |
19 | 18 | eqeq1d 2726 | . . 3 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0)) |
20 | 19 | rspcev 3604 | . 2 ⊢ ((〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ∧ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
21 | 7, 17, 20 | syl2anc 583 | 1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 〈cop 4627 ‘cfv 6534 (class class class)co 7402 1st c1st 7967 Rcnr 10857 0Rc0r 10858 -1Rcm1r 10860 +R cplr 10861 ·R cmr 10862 ℝcr 11106 0cc0 11107 + caddc 11110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8700 df-ec 8702 df-qs 8706 df-ni 10864 df-pli 10865 df-mi 10866 df-lti 10867 df-plpq 10900 df-mpq 10901 df-ltpq 10902 df-enq 10903 df-nq 10904 df-erq 10905 df-plq 10906 df-mq 10907 df-1nq 10908 df-rq 10909 df-ltnq 10910 df-np 10973 df-1p 10974 df-plp 10975 df-mp 10976 df-ltp 10977 df-enr 11047 df-nr 11048 df-plr 11049 df-mr 11050 df-0r 11052 df-1r 11053 df-m1r 11054 df-c 11113 df-0 11114 df-r 11117 df-add 11118 |
This theorem is referenced by: (None) |
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