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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-mul02 | Structured version Visualization version GIF version |
Description: mul02 11437 without ax-mulcom 11217. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11217 for an outline. (Contributed by SN, 30-Jun-2024.) |
Ref | Expression |
---|---|
sn-mul02 | ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 11256 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
2 | 0cnd 11252 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 0 ∈ ℂ) | |
3 | recn 11243 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ) |
5 | ax-icn 11212 | . . . . . . . 8 ⊢ i ∈ ℂ | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ) |
7 | recn 11243 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | 7 | adantl 481 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
9 | 6, 8 | mulcld 11279 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ) |
10 | 2, 4, 9 | adddid 11283 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) |
11 | remul02 42412 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (0 · 𝑥) = 0) | |
12 | 11 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · 𝑥) = 0) |
13 | sn-0tie0 42446 | . . . . . . . . 9 ⊢ (0 · i) = 0 | |
14 | 13 | oveq1i 7441 | . . . . . . . 8 ⊢ ((0 · i) · 𝑦) = (0 · 𝑦) |
15 | 2, 6, 8 | mulassd 11282 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · i) · 𝑦) = (0 · (i · 𝑦))) |
16 | remul02 42412 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (0 · 𝑦) = 0) | |
17 | 16 | adantl 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · 𝑦) = 0) |
18 | 14, 15, 17 | 3eqtr3a 2799 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (i · 𝑦)) = 0) |
19 | 12, 18 | oveq12d 7449 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · 𝑥) + (0 · (i · 𝑦))) = (0 + 0)) |
20 | sn-00id 42408 | . . . . . 6 ⊢ (0 + 0) = 0 | |
21 | 19, 20 | eqtrdi 2791 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · 𝑥) + (0 · (i · 𝑦))) = 0) |
22 | 10, 21 | eqtrd 2775 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = 0) |
23 | oveq2 7439 | . . . . 5 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = (0 · (𝑥 + (i · 𝑦)))) | |
24 | 23 | eqeq1d 2737 | . . . 4 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → ((0 · 𝐴) = 0 ↔ (0 · (𝑥 + (i · 𝑦))) = 0)) |
25 | 22, 24 | syl5ibrcom 247 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = 0)) |
26 | 25 | rexlimivv 3199 | . 2 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = 0) |
27 | 1, 26 | syl 17 | 1 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 ici 11155 + caddc 11156 · cmul 11158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-2 12327 df-3 12328 df-resub 42373 |
This theorem is referenced by: (None) |
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