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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 3149 | . . 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 19262 | . . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
7 | oveq2 7143 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 )) | |
8 | 7 | eqeq1d 2800 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍)) |
9 | 8 | rspcv 3566 | . . . 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 19269 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 ) |
15 | 3, 12, 14 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 ) |
16 | 11, 15 | eqtr3d 2835 | 1 ⊢ (𝜑 → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 0gc0g 16705 SRingcsrg 19248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-cmn 18900 df-srg 19249 |
This theorem is referenced by: (None) |
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