Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringcvalALTV | Structured version Visualization version Unicode version |
Description: Value of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringcvalALTV.c | RingCatALTV |
ringcvalALTV.u | |
ringcvalALTV.b | |
ringcvalALTV.h | RingHom |
ringcvalALTV.o | RingHom RingHom |
Ref | Expression |
---|---|
ringcvalALTV | comp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcvalALTV.c | . 2 RingCatALTV | |
2 | df-ringcALTV 42006 | . . . 4 RingCatALTV RingHom comp RingHom RingHom | |
3 | 2 | a1i 11 | . . 3 RingCatALTV RingHom comp RingHom RingHom |
4 | vex 3203 | . . . . . 6 | |
5 | 4 | inex1 4799 | . . . . 5 |
6 | 5 | a1i 11 | . . . 4 |
7 | ineq1 3807 | . . . . . 6 | |
8 | 7 | adantl 482 | . . . . 5 |
9 | ringcvalALTV.b | . . . . . 6 | |
10 | 9 | adantr 481 | . . . . 5 |
11 | 8, 10 | eqtr4d 2659 | . . . 4 |
12 | simpr 477 | . . . . . 6 | |
13 | 12 | opeq2d 4409 | . . . . 5 |
14 | eqidd 2623 | . . . . . . . 8 RingHom RingHom | |
15 | 12, 12, 14 | mpt2eq123dv 6717 | . . . . . . 7 RingHom RingHom |
16 | ringcvalALTV.h | . . . . . . . 8 RingHom | |
17 | 16 | ad2antrr 762 | . . . . . . 7 RingHom |
18 | 15, 17 | eqtr4d 2659 | . . . . . 6 RingHom |
19 | 18 | opeq2d 4409 | . . . . 5 RingHom |
20 | 12 | sqxpeqd 5141 | . . . . . . . 8 |
21 | eqidd 2623 | . . . . . . . 8 RingHom RingHom RingHom RingHom | |
22 | 20, 12, 21 | mpt2eq123dv 6717 | . . . . . . 7 RingHom RingHom RingHom RingHom |
23 | ringcvalALTV.o | . . . . . . . 8 RingHom RingHom | |
24 | 23 | ad2antrr 762 | . . . . . . 7 RingHom RingHom |
25 | 22, 24 | eqtr4d 2659 | . . . . . 6 RingHom RingHom |
26 | 25 | opeq2d 4409 | . . . . 5 comp RingHom RingHom comp |
27 | 13, 19, 26 | tpeq123d 4283 | . . . 4 RingHom comp RingHom RingHom comp |
28 | 6, 11, 27 | csbied2 3561 | . . 3 RingHom comp RingHom RingHom comp |
29 | ringcvalALTV.u | . . . 4 | |
30 | elex 3212 | . . . 4 | |
31 | 29, 30 | syl 17 | . . 3 |
32 | tpex 6957 | . . . 4 comp | |
33 | 32 | a1i 11 | . . 3 comp |
34 | 3, 28, 31, 33 | fvmptd 6288 | . 2 RingCatALTV comp |
35 | 1, 34 | syl5eq 2668 | 1 comp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 csb 3533 cin 3573 ctp 4181 cop 4183 cmpt 4729 cxp 5112 ccom 5118 cfv 5888 (class class class)co 6650 cmpt2 6652 c1st 7166 c2nd 7167 cnx 15854 cbs 15857 chom 15952 compcco 15953 crg 18547 RingHom crh 18712 RingCatALTVcringcALTV 42004 |
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-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-tp 4182 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-iota 5851 df-fun 5890 df-fv 5896 df-oprab 6654 df-mpt2 6655 df-ringcALTV 42006 |
This theorem is referenced by: ringcbasALTV 42046 ringchomfvalALTV 42047 ringccofvalALTV 42050 |
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