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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 3144 | . . 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 20218 | . . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
7 | oveq2 7439 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 )) | |
8 | 7 | eqeq1d 2737 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍)) |
9 | 8 | rspcv 3618 | . . . 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 20225 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 ) |
15 | 3, 12, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 ) |
16 | 11, 15 | eqtr3d 2777 | 1 ⊢ (𝜑 → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 0gc0g 17486 SRingcsrg 20204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cmn 19815 df-srg 20205 |
This theorem is referenced by: (None) |
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