Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcval | Structured version Visualization version Unicode version |
Description: Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
rngcval.c | RngCat |
rngcval.u | |
rngcval.b | Rng |
rngcval.h | RngHomo |
Ref | Expression |
---|---|
rngcval | ExtStrCat cat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcval.c | . 2 RngCat | |
2 | df-rngc 41959 | . . . 4 RngCat ExtStrCat cat RngHomo Rng Rng | |
3 | 2 | a1i 11 | . . 3 RngCat ExtStrCat cat RngHomo Rng Rng |
4 | fveq2 6191 | . . . . 5 ExtStrCat ExtStrCat | |
5 | 4 | adantl 482 | . . . 4 ExtStrCat ExtStrCat |
6 | ineq1 3807 | . . . . . . . 8 Rng Rng | |
7 | 6 | sqxpeqd 5141 | . . . . . . 7 Rng Rng Rng Rng |
8 | rngcval.b | . . . . . . . . 9 Rng | |
9 | 8 | sqxpeqd 5141 | . . . . . . . 8 Rng Rng |
10 | 9 | eqcomd 2628 | . . . . . . 7 Rng Rng |
11 | 7, 10 | sylan9eqr 2678 | . . . . . 6 Rng Rng |
12 | 11 | reseq2d 5396 | . . . . 5 RngHomo Rng Rng RngHomo |
13 | rngcval.h | . . . . . . 7 RngHomo | |
14 | 13 | eqcomd 2628 | . . . . . 6 RngHomo |
15 | 14 | adantr 481 | . . . . 5 RngHomo |
16 | 12, 15 | eqtrd 2656 | . . . 4 RngHomo Rng Rng |
17 | 5, 16 | oveq12d 6668 | . . 3 ExtStrCat cat RngHomo Rng Rng ExtStrCat cat |
18 | rngcval.u | . . . 4 | |
19 | elex 3212 | . . . 4 | |
20 | 18, 19 | syl 17 | . . 3 |
21 | ovexd 6680 | . . 3 ExtStrCat cat | |
22 | 3, 17, 20, 21 | fvmptd 6288 | . 2 RngCat ExtStrCat cat |
23 | 1, 22 | syl5eq 2668 | 1 ExtStrCat cat |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cin 3573 cmpt 4729 cxp 5112 cres 5116 cfv 5888 (class class class)co 6650 cat cresc 16468 ExtStrCatcestrc 16762 Rngcrng 41874 RngHomo crngh 41885 RngCatcrngc 41957 |
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-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 |
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-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-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-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-rngc 41959 |
This theorem is referenced by: rngcbas 41965 rngchomfval 41966 rngccofval 41970 dfrngc2 41972 rngccat 41978 rngcid 41979 rngcifuestrc 41997 funcrngcsetc 41998 |
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