![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cnegex2 | GIF version |
Description: Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
cnegex2 | ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex 8164 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) | |
2 | addcom 8123 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑥) = (𝑥 + 𝐴)) | |
3 | 2 | eqeq1d 2198 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 ↔ (𝑥 + 𝐴) = 0)) |
4 | 3 | rexbidva 2487 | . 2 ⊢ (𝐴 ∈ ℂ → (∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 ↔ ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)) |
5 | 1, 4 | mpbid 147 | 1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 (class class class)co 5895 ℂcc 7838 0cc0 7840 + caddc 7843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-resscn 7932 ax-1cn 7933 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5898 |
This theorem is referenced by: addcan 8166 |
Copyright terms: Public domain | W3C validator |