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Mirrors > Home > ILE Home > Th. List > prodeq2w | Unicode version |
Description: Equality theorem for product, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodeq2w |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . . . . . . . . . . . 13 | |
2 | ifeq1 3477 | . . . . . . . . . . . . . . 15 | |
3 | 2 | alimi 1431 | . . . . . . . . . . . . . 14 |
4 | alral 2478 | . . . . . . . . . . . . . 14 | |
5 | 3, 4 | syl 14 | . . . . . . . . . . . . 13 |
6 | mpteq12 4011 | . . . . . . . . . . . . 13 | |
7 | 1, 5, 6 | sylancr 410 | . . . . . . . . . . . 12 |
8 | 7 | seqeq3d 10229 | . . . . . . . . . . 11 |
9 | 8 | breq1d 3939 | . . . . . . . . . 10 |
10 | 9 | anbi2d 459 | . . . . . . . . 9 # # |
11 | 10 | exbidv 1797 | . . . . . . . 8 # # |
12 | 11 | rexbidv 2438 | . . . . . . 7 # # |
13 | 7 | seqeq3d 10229 | . . . . . . . 8 |
14 | 13 | breq1d 3939 | . . . . . . 7 |
15 | 12, 14 | anbi12d 464 | . . . . . 6 # # |
16 | 15 | anbi2d 459 | . . . . 5 DECID # DECID # |
17 | 16 | rexbidv 2438 | . . . 4 DECID # DECID # |
18 | csbeq2 3026 | . . . . . . . . . . . 12 | |
19 | 18 | ifeq1d 3489 | . . . . . . . . . . 11 |
20 | 19 | mpteq2dv 4019 | . . . . . . . . . 10 |
21 | 20 | seqeq3d 10229 | . . . . . . . . 9 |
22 | 21 | fveq1d 5423 | . . . . . . . 8 |
23 | 22 | eqeq2d 2151 | . . . . . . 7 |
24 | 23 | anbi2d 459 | . . . . . 6 |
25 | 24 | exbidv 1797 | . . . . 5 |
26 | 25 | rexbidv 2438 | . . . 4 |
27 | 17, 26 | orbi12d 782 | . . 3 DECID # DECID # |
28 | 27 | iotabidv 5109 | . 2 DECID # DECID # |
29 | df-proddc 11323 | . 2 DECID # | |
30 | df-proddc 11323 | . 2 DECID # | |
31 | 28, 29, 30 | 3eqtr4g 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 DECID wdc 819 wal 1329 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 csb 3003 wss 3071 cif 3474 class class class wbr 3929 cmpt 3989 cio 5086 wf1o 5122 cfv 5123 (class class class)co 5774 cc0 7623 c1 7624 cmul 7628 cle 7804 # cap 8346 cn 8723 cz 9057 cuz 9329 cfz 9793 cseq 10221 cli 11050 cprod 11322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-seqfrec 10222 df-proddc 11323 |
This theorem is referenced by: (None) |
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