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| Mirrors > Home > MPE Home > Th. List > srgisid | Structured version Visualization version GIF version | ||
| Description: In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.) | 
| Ref | Expression | 
|---|---|
| srgz.b | ⊢ 𝐵 = (Base‘𝑅) | 
| srgz.t | ⊢ · = (.r‘𝑅) | 
| srgz.z | ⊢ 0 = (0g‘𝑅) | 
| srgisid.1 | ⊢ (𝜑 → 𝑅 ∈ SRing) | 
| srgisid.2 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) | 
| srgisid.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍) | 
| Ref | Expression | 
|---|---|
| srgisid | ⊢ (𝜑 → 𝑍 = 0 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | srgisid.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍) | |
| 2 | 1 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍) | 
| 3 | srgisid.1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing) | |
| 4 | srgz.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 5 | srgz.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
| 6 | 4, 5 | srg0cl 20198 | . . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) | 
| 7 | oveq2 7440 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 )) | |
| 8 | 7 | eqeq1d 2738 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍)) | 
| 9 | 8 | rspcv 3617 | . . . 4 ⊢ ( 0 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍)) | 
| 10 | 3, 6, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍)) | 
| 11 | 2, 10 | mpd 15 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 𝑍) | 
| 12 | srgisid.2 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 13 | srgz.t | . . . 4 ⊢ · = (.r‘𝑅) | |
| 14 | 4, 13, 5 | srgrz 20205 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 ) | 
| 15 | 3, 12, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 ) | 
| 16 | 11, 15 | eqtr3d 2778 | 1 ⊢ (𝜑 → 𝑍 = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 .rcmulr 17299 0gc0g 17485 SRingcsrg 20184 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-cmn 19801 df-srg 20185 | 
| This theorem is referenced by: (None) | 
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