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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 3175 | . . 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 18880 | . . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
7 | oveq2 6918 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 )) | |
8 | 7 | eqeq1d 2827 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍)) |
9 | 8 | rspcv 3522 | . . . 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 18887 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 ) |
15 | 3, 12, 14 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 ) |
16 | 11, 15 | eqtr3d 2863 | 1 ⊢ (𝜑 → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 .rcmulr 16313 0gc0g 16460 SRingcsrg 18866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-riota 6871 df-ov 6913 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-cmn 18555 df-srg 18867 |
This theorem is referenced by: (None) |
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