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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-addid1 | Structured version Visualization version GIF version |
Description: addid1 10843 without ax-mulcom 10624. (Contributed by SN, 5-May-2024.) |
Ref | Expression |
---|---|
sn-addid1 | ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sn-negex2 39882 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) | |
2 | simprr 773 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (𝑥 + 𝐴) = 0) | |
3 | 2 | oveq1d 7158 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + 𝐴) + 0) = (0 + 0)) |
4 | sn-00id 39866 | . . . . 5 ⊢ (0 + 0) = 0 | |
5 | 3, 4 | eqtrdi 2810 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + 𝐴) + 0) = 0) |
6 | simprl 771 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → 𝑥 ∈ ℂ) | |
7 | simpl 487 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → 𝐴 ∈ ℂ) | |
8 | 0cnd 10657 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → 0 ∈ ℂ) | |
9 | 6, 7, 8 | addassd 10686 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + 𝐴) + 0) = (𝑥 + (𝐴 + 0))) |
10 | 2, 5, 9 | 3eqtr2rd 2801 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (𝑥 + (𝐴 + 0)) = (𝑥 + 𝐴)) |
11 | 7, 8 | addcld 10683 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (𝐴 + 0) ∈ ℂ) |
12 | 6, 11, 7 | sn-addcand 39883 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + (𝐴 + 0)) = (𝑥 + 𝐴) ↔ (𝐴 + 0) = 𝐴)) |
13 | 10, 12 | mpbid 235 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (𝐴 + 0) = 𝐴) |
14 | 1, 13 | rexlimddv 3213 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 (class class class)co 7143 ℂcc 10558 0cc0 10560 + caddc 10563 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-ltxr 10703 df-2 11722 df-3 11723 df-resub 39831 |
This theorem is referenced by: sn-addcan2d 39885 sn-addid0 39888 |
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