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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 10007. (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 9953 | . . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | |
2 | 1 | simplbi 476 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R) |
3 | m1r 9903 | . . . 4 ⊢ -1R ∈ R | |
4 | mulclsr 9905 | . . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 694 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R) |
6 | opelreal 9951 | . . 3 ⊢ (〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R) | |
7 | 5, 6 | sylibr 224 | . 2 ⊢ (𝐴 ∈ ℝ → 〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ) |
8 | 1 | simprbi 480 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
9 | 8 | oveq1d 6665 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) |
10 | addresr 9959 | . . . 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 693 | . . 3 ⊢ (𝐴 ∈ ℝ → (〈(1st ‘𝐴), 0R〉 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉) |
12 | pn0sr 9922 | . . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R) | |
13 | 12 | opeq1d 4408 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ R → 〈((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R〉 = 〈0R, 0R〉) |
14 | df-0 9943 | . . . . 5 ⊢ 0 = 〈0R, 0R〉 | |
15 | 13, 14 | syl6eqr 2674 | . . . 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 2660 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) |
18 | oveq2 6658 | . . . 4 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → (𝐴 + 𝑥) = (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉)) | |
19 | 18 | eqeq1d 2624 | . . 3 ⊢ (𝑥 = 〈((1st ‘𝐴) ·R -1R), 0R〉 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0)) |
20 | 19 | rspcev 3309 | . 2 ⊢ ((〈((1st ‘𝐴) ·R -1R), 0R〉 ∈ ℝ ∧ (𝐴 + 〈((1st ‘𝐴) ·R -1R), 0R〉) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
21 | 7, 17, 20 | syl2anc 693 | 1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 〈cop 4183 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 Rcnr 9687 0Rc0r 9688 -1Rcm1r 9690 +R cplr 9691 ·R cmr 9692 ℝcr 9935 0cc0 9936 + caddc 9939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 df-1p 9804 df-plp 9805 df-mp 9806 df-ltp 9807 df-enr 9877 df-nr 9878 df-plr 9879 df-mr 9880 df-0r 9882 df-1r 9883 df-m1r 9884 df-c 9942 df-0 9943 df-r 9946 df-add 9947 |
This theorem is referenced by: (None) |
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