Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10942. (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 10888 | . . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | |
2 | 1 | simplbi 498 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R) |
3 | m1r 10838 | . . . 4 ⊢ -1R ∈ R | |
4 | mulclsr 10840 | . . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 586 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R) |
6 | opelreal 10886 | . . 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 497 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
9 | 8 | oveq1d 7290 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) |
10 | addresr 10894 | . . . 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 584 | . . 3 ⊢ (𝐴 ∈ ℝ → (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉) |
12 | pn0sr 10857 | . . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R) | |
13 | 12 | opeq1d 4810 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ R → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 〈0R, 0R〉) |
14 | df-0 10878 | . . . . 5 ⊢ 0 = 〈0R, 0R〉 | |
15 | 13, 14 | eqtr4di 2796 | . . . 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 2782 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) |
18 | oveq2 7283 | . . . 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 3561 | . 2 ⊢ ((〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ∧ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
21 | 7, 17, 20 | syl2anc 584 | 1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 〈cop 4567 ‘cfv 6433 (class class class)co 7275 1st c1st 7829 Rcnr 10621 0Rc0r 10622 -1Rcm1r 10624 +R cplr 10625 ·R cmr 10626 ℝcr 10870 0cc0 10871 + caddc 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-omul 8302 df-er 8498 df-ec 8500 df-qs 8504 df-ni 10628 df-pli 10629 df-mi 10630 df-lti 10631 df-plpq 10664 df-mpq 10665 df-ltpq 10666 df-enq 10667 df-nq 10668 df-erq 10669 df-plq 10670 df-mq 10671 df-1nq 10672 df-rq 10673 df-ltnq 10674 df-np 10737 df-1p 10738 df-plp 10739 df-mp 10740 df-ltp 10741 df-enr 10811 df-nr 10812 df-plr 10813 df-mr 10814 df-0r 10816 df-1r 10817 df-m1r 10818 df-c 10877 df-0 10878 df-r 10881 df-add 10882 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |