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Mirrors > Home > MPE Home > Th. List > Mathboxes > cmpidelt | Structured version Visualization version GIF version |
Description: A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cmpidelt.1 | ⊢ 𝑋 = ran 𝐺 |
cmpidelt.2 | ⊢ 𝑈 = (GId‘𝐺) |
Ref | Expression |
---|---|
cmpidelt | ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmpidelt.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
2 | cmpidelt.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐺) | |
3 | 1, 2 | idrval 33656 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
4 | 3 | eqcomd 2628 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈) |
5 | 1, 2 | iorlid 33657 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) |
6 | 1 | exidu1 33655 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) |
7 | oveq1 6657 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥)) | |
8 | 7 | eqeq1d 2624 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥)) |
9 | oveq2 6658 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑥𝐺𝑢) = (𝑥𝐺𝑈)) | |
10 | 9 | eqeq1d 2624 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑥𝐺𝑢) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥)) |
11 | 8, 10 | anbi12d 747 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))) |
12 | 11 | ralbidv 2986 | . . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))) |
13 | 12 | riota2 6633 | . . . 4 ⊢ ((𝑈 ∈ 𝑋 ∧ ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈)) |
14 | 5, 6, 13 | syl2anc 693 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈)) |
15 | 4, 14 | mpbird 247 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)) |
16 | oveq2 6658 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴)) | |
17 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
18 | 16, 17 | eqeq12d 2637 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴)) |
19 | oveq1 6657 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈)) | |
20 | 19, 17 | eqeq12d 2637 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴)) |
21 | 18, 20 | anbi12d 747 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))) |
22 | 21 | rspccva 3308 | . 2 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
23 | 15, 22 | sylan 488 | 1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃!wreu 2914 ∩ cin 3573 ran crn 5115 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 GIdcgi 27344 ExId cexid 33643 Magmacmagm 33647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-riota 6611 df-ov 6653 df-gid 27348 df-exid 33644 df-mgmOLD 33648 |
This theorem is referenced by: exidreslem 33676 rngoidmlem 33735 |
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