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Mirrors > Home > MPE Home > Th. List > Mathboxes > iorlid | Structured version Visualization version GIF version |
Description: A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
iorlid.1 | ⊢ 𝑋 = ran 𝐺 |
iorlid.2 | ⊢ 𝑈 = (GId‘𝐺) |
Ref | Expression |
---|---|
iorlid | ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iorlid.1 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
2 | iorlid.2 | . . 3 ⊢ 𝑈 = (GId‘𝐺) | |
3 | 1, 2 | idrval 36366 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
4 | 1 | exidu1 36365 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) |
5 | riotacl 7335 | . . 3 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋) |
7 | 3, 6 | eqeltrd 2834 | 1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃!wreu 3350 ∩ cin 3913 ran crn 5638 ‘cfv 6500 ℩crio 7316 (class class class)co 7361 GIdcgi 29481 ExId cexid 36353 Magmacmagm 36357 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-riota 7317 df-ov 7364 df-gid 29485 df-exid 36354 df-mgmOLD 36358 |
This theorem is referenced by: cmpidelt 36368 rngo1cl 36448 |
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