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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-addlid | Structured version Visualization version GIF version |
Description: addlid 11427 without ax-mulcom 11202. (Contributed by SN, 23-Jan-2024.) |
Ref | Expression |
---|---|
sn-addlid | ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 11241 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
2 | 0cnd 11237 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → 0 ∈ ℂ) | |
3 | simp2l 1197 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → 𝑥 ∈ ℝ) | |
4 | 3 | recnd 11272 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → 𝑥 ∈ ℂ) |
5 | ax-icn 11197 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → i ∈ ℂ) |
7 | simp2r 1198 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → 𝑦 ∈ ℝ) | |
8 | 7 | recnd 11272 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → 𝑦 ∈ ℂ) |
9 | 6, 8 | mulcld 11264 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → (i · 𝑦) ∈ ℂ) |
10 | 2, 4, 9 | addassd 11266 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → ((0 + 𝑥) + (i · 𝑦)) = (0 + (𝑥 + (i · 𝑦)))) |
11 | readdlid 41958 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (0 + 𝑥) = 𝑥) | |
12 | 11 | adantr 480 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 + 𝑥) = 𝑥) |
13 | 12 | 3ad2ant2 1132 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → (0 + 𝑥) = 𝑥) |
14 | 13 | oveq1d 7435 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → ((0 + 𝑥) + (i · 𝑦)) = (𝑥 + (i · 𝑦))) |
15 | 10, 14 | eqtr3d 2770 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → (0 + (𝑥 + (i · 𝑦))) = (𝑥 + (i · 𝑦))) |
16 | simp3 1136 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → 𝐴 = (𝑥 + (i · 𝑦))) | |
17 | 16 | oveq2d 7436 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → (0 + 𝐴) = (0 + (𝑥 + (i · 𝑦)))) |
18 | 15, 17, 16 | 3eqtr4d 2778 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐴 = (𝑥 + (i · 𝑦))) → (0 + 𝐴) = 𝐴) |
19 | 18 | 3exp 1117 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (0 + 𝐴) = 𝐴))) |
20 | 19 | rexlimdvv 3207 | . 2 ⊢ (𝐴 ∈ ℂ → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)) → (0 + 𝐴) = 𝐴)) |
21 | 1, 20 | mpd 15 | 1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 (class class class)co 7420 ℂcc 11136 ℝcr 11137 0cc0 11138 ici 11140 + caddc 11141 · cmul 11143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-resub 41921 |
This theorem is referenced by: sn-it0e0 41970 sn-negex12 41971 sn-addcand 41974 sn-subeu 41981 sn-0tie0 41994 cnreeu 42023 |
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