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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-subeu | Structured version Visualization version GIF version |
Description: negeu 11033 without ax-mulcom 10758 and complex number version of resubeu 40009. (Contributed by SN, 5-May-2024.) |
Ref | Expression |
---|---|
sn-subeu | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sn-negex 40048 | . . 3 ⊢ (𝐴 ∈ ℂ → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0) |
3 | simprl 771 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → 𝑦 ∈ ℂ) | |
4 | simplr 769 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → 𝐵 ∈ ℂ) | |
5 | 3, 4 | addcld 10817 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → (𝑦 + 𝐵) ∈ ℂ) |
6 | simplrr 778 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑦) = 0) | |
7 | 6 | oveq1d 7206 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (0 + 𝐵)) |
8 | simplll 775 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) | |
9 | simplrl 777 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑦 ∈ ℂ) | |
10 | simpllr 776 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) | |
11 | 8, 9, 10 | addassd 10820 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (𝐴 + (𝑦 + 𝐵))) |
12 | sn-addid2 40036 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵) | |
13 | 10, 12 | syl 17 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (0 + 𝐵) = 𝐵) |
14 | 7, 11, 13 | 3eqtr3rd 2780 | . . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 = (𝐴 + (𝑦 + 𝐵))) |
15 | 14 | eqeq2d 2747 | . . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵 ↔ (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵)))) |
16 | simpr 488 | . . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
17 | 9, 10 | addcld 10817 | . . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝑦 + 𝐵) ∈ ℂ) |
18 | 8, 16, 17 | sn-addcand 40050 | . . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵)) ↔ 𝑥 = (𝑦 + 𝐵))) |
19 | 15, 18 | bitrd 282 | . . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵 ↔ 𝑥 = (𝑦 + 𝐵))) |
20 | 19 | ralrimiva 3095 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵 ↔ 𝑥 = (𝑦 + 𝐵))) |
21 | reu6i 3630 | . . 3 ⊢ (((𝑦 + 𝐵) ∈ ℂ ∧ ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵 ↔ 𝑥 = (𝑦 + 𝐵))) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵) | |
22 | 5, 20, 21 | syl2anc 587 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵) |
23 | 2, 22 | rexlimddv 3200 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ∃!wreu 3053 (class class class)co 7191 ℂcc 10692 0cc0 10694 + caddc 10697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-2 11858 df-3 11859 df-resub 39998 |
This theorem is referenced by: sn-subcl 40058 resubeqsub 40060 addinvcom 40062 |
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