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Mirrors > Home > MPE Home > Th. List > issubrg | Structured version Visualization version Unicode version |
Description: The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.) |
Ref | Expression |
---|---|
issubrg.b | |
issubrg.i |
Ref | Expression |
---|---|
issubrg | SubRing ↾s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-subrg 18778 | . . 3 SubRing ↾s | |
2 | 1 | mptrcl 6289 | . 2 SubRing |
3 | simpll 790 | . 2 ↾s | |
4 | fveq2 6191 | . . . . . . . 8 | |
5 | issubrg.b | . . . . . . . 8 | |
6 | 4, 5 | syl6eqr 2674 | . . . . . . 7 |
7 | 6 | pweqd 4163 | . . . . . 6 |
8 | oveq1 6657 | . . . . . . . 8 ↾s ↾s | |
9 | 8 | eleq1d 2686 | . . . . . . 7 ↾s ↾s |
10 | fveq2 6191 | . . . . . . . . 9 | |
11 | issubrg.i | . . . . . . . . 9 | |
12 | 10, 11 | syl6eqr 2674 | . . . . . . . 8 |
13 | 12 | eleq1d 2686 | . . . . . . 7 |
14 | 9, 13 | anbi12d 747 | . . . . . 6 ↾s ↾s |
15 | 7, 14 | rabeqbidv 3195 | . . . . 5 ↾s ↾s |
16 | fvex 6201 | . . . . . . . 8 | |
17 | 5, 16 | eqeltri 2697 | . . . . . . 7 |
18 | 17 | pwex 4848 | . . . . . 6 |
19 | 18 | rabex 4813 | . . . . 5 ↾s |
20 | 15, 1, 19 | fvmpt 6282 | . . . 4 SubRing ↾s |
21 | 20 | eleq2d 2687 | . . 3 SubRing ↾s |
22 | oveq2 6658 | . . . . . . . 8 ↾s ↾s | |
23 | 22 | eleq1d 2686 | . . . . . . 7 ↾s ↾s |
24 | eleq2 2690 | . . . . . . 7 | |
25 | 23, 24 | anbi12d 747 | . . . . . 6 ↾s ↾s |
26 | 25 | elrab 3363 | . . . . 5 ↾s ↾s |
27 | 17 | elpw2 4828 | . . . . . 6 |
28 | 27 | anbi1i 731 | . . . . 5 ↾s ↾s |
29 | an12 838 | . . . . 5 ↾s ↾s | |
30 | 26, 28, 29 | 3bitri 286 | . . . 4 ↾s ↾s |
31 | ibar 525 | . . . . 5 ↾s ↾s | |
32 | 31 | anbi1d 741 | . . . 4 ↾s ↾s |
33 | 30, 32 | syl5bb 272 | . . 3 ↾s ↾s |
34 | 21, 33 | bitrd 268 | . 2 SubRing ↾s |
35 | 2, 3, 34 | pm5.21nii 368 | 1 SubRing ↾s |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 wss 3574 cpw 4158 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cur 18501 crg 18547 SubRingcsubrg 18776 |
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-pow 4843 ax-pr 4906 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-subrg 18778 |
This theorem is referenced by: subrgss 18781 subrgid 18782 subrgring 18783 subrgrcl 18785 subrg1cl 18788 issubrg2 18800 subsubrg 18806 subrgpropd 18814 issubassa 19324 subrgpsr 19419 cphsubrglem 22977 |
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