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Mirrors > Home > MPE Home > Th. List > ismgmid2 | Structured version Visualization version GIF version |
Description: Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ismgmid.b | ⊢ 𝐵 = (Base‘𝐺) |
ismgmid.o | ⊢ 0 = (0g‘𝐺) |
ismgmid.p | ⊢ + = (+g‘𝐺) |
ismgmid2.u | ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
ismgmid2.l | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥) |
ismgmid2.r | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥) |
Ref | Expression |
---|---|
ismgmid2 | ⊢ (𝜑 → 𝑈 = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismgmid2.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵) | |
2 | ismgmid2.l | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥) | |
3 | ismgmid2.r | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥) | |
4 | 2, 3 | jca 513 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) |
5 | 4 | ralrimiva 3142 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) |
6 | ismgmid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
7 | ismgmid.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
8 | ismgmid.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | oveq1 7357 | . . . . . . . 8 ⊢ (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥)) | |
10 | 9 | eqeq1d 2740 | . . . . . . 7 ⊢ (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥)) |
11 | 10 | ovanraleqv 7374 | . . . . . 6 ⊢ (𝑒 = 𝑈 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))) |
12 | 11 | rspcev 3580 | . . . . 5 ⊢ ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
13 | 1, 5, 12 | syl2anc 585 | . . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
14 | 6, 7, 8, 13 | ismgmid 18456 | . . 3 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈)) |
15 | 1, 5, 14 | mpbi2and 711 | . 2 ⊢ (𝜑 → 0 = 𝑈) |
16 | 15 | eqcomd 2744 | 1 ⊢ (𝜑 → 𝑈 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ∃wrex 3072 ‘cfv 6492 (class class class)co 7350 Basecbs 17019 +gcplusg 17069 0gc0g 17257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6444 df-fun 6494 df-fv 6500 df-riota 7306 df-ov 7353 df-0g 17259 |
This theorem is referenced by: lidrididd 18461 grpidd 18462 submnd0 18521 mnd1id 18534 frmd0 18606 efmndid 18634 pwmndid 18682 mhmid 18803 cnaddid 19583 ringidss 19928 xrs10 20765 idlsrg0g 32044 |
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