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Mirrors > Home > ILE Home > Th. List > xpima1 | Unicode version |
Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Ref | Expression |
---|---|
xpima1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4552 | . . 3 | |
2 | df-res 4551 | . . . 4 | |
3 | 2 | rneqi 4767 | . . 3 |
4 | inxp 4673 | . . . 4 | |
5 | 4 | rneqi 4767 | . . 3 |
6 | 1, 3, 5 | 3eqtri 2164 | . 2 |
7 | xpeq1 4553 | . . . 4 | |
8 | 0xp 4619 | . . . 4 | |
9 | 7, 8 | syl6eq 2188 | . . 3 |
10 | rneq 4766 | . . . 4 | |
11 | rn0 4795 | . . . 4 | |
12 | 10, 11 | syl6eq 2188 | . . 3 |
13 | 9, 12 | syl 14 | . 2 |
14 | 6, 13 | syl5eq 2184 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 cvv 2686 cin 3070 c0 3363 cxp 4537 crn 4540 cres 4541 cima 4542 |
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-in1 603 ax-in2 604 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 |
This theorem is referenced by: (None) |
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