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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-mul02 | Structured version Visualization version GIF version | ||
| Description: mul02 11439 without ax-mulcom 11219. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11219 for an outline. (Contributed by SN, 30-Jun-2024.) |
| Ref | Expression |
|---|---|
| sn-mul02 | ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnre 11258 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
| 2 | 0cnd 11254 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 0 ∈ ℂ) | |
| 3 | recn 11245 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
| 4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ) |
| 5 | ax-icn 11214 | . . . . . . . 8 ⊢ i ∈ ℂ | |
| 6 | 5 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ) |
| 7 | recn 11245 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
| 8 | 7 | adantl 481 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
| 9 | 6, 8 | mulcld 11281 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ) |
| 10 | 2, 4, 9 | adddid 11285 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) |
| 11 | remul02 42435 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (0 · 𝑥) = 0) | |
| 12 | 11 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · 𝑥) = 0) |
| 13 | sn-0tie0 42469 | . . . . . . . . 9 ⊢ (0 · i) = 0 | |
| 14 | 13 | oveq1i 7441 | . . . . . . . 8 ⊢ ((0 · i) · 𝑦) = (0 · 𝑦) |
| 15 | 2, 6, 8 | mulassd 11284 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · i) · 𝑦) = (0 · (i · 𝑦))) |
| 16 | remul02 42435 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (0 · 𝑦) = 0) | |
| 17 | 16 | adantl 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · 𝑦) = 0) |
| 18 | 14, 15, 17 | 3eqtr3a 2801 | . . . . . . 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 42431 | . . . . . 6 ⊢ (0 + 0) = 0 | |
| 21 | 19, 20 | eqtrdi 2793 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · 𝑥) + (0 · (i · 𝑦))) = 0) |
| 22 | 10, 21 | eqtrd 2777 | . . . 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 2739 | . . . 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 3201 | . 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 1540 ∈ wcel 2108 ∃wrex 3070 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 ici 11157 + caddc 11158 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-2 12329 df-3 12330 df-resub 42396 |
| This theorem is referenced by: (None) |
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