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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brpprod3a | Structured version Visualization version Unicode version |
Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brpprod3.1 |
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brpprod3.2 |
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brpprod3.3 |
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Ref | Expression |
---|---|
brpprod3a |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pprodss4v 30700 |
. . . . . . 7
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2 | 1 | brel 4902 |
. . . . . 6
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3 | 2 | simprd 469 |
. . . . 5
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4 | elvv 4912 |
. . . . 5
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5 | 3, 4 | sylib 201 |
. . . 4
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6 | 5 | pm4.71ri 643 |
. . 3
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7 | 19.41vv 1842 |
. . 3
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8 | 6, 7 | bitr4i 260 |
. 2
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9 | breq2 4420 |
. . . 4
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10 | 9 | pm5.32i 647 |
. . 3
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11 | 10 | 2exbii 1730 |
. 2
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12 | brpprod3.1 |
. . . . . 6
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13 | brpprod3.2 |
. . . . . 6
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14 | vex 3060 |
. . . . . 6
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15 | vex 3060 |
. . . . . 6
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16 | 12, 13, 14, 15 | brpprod 30701 |
. . . . 5
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17 | 16 | anbi2i 705 |
. . . 4
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18 | 3anass 995 |
. . . 4
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19 | 17, 18 | bitr4i 260 |
. . 3
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20 | 19 | 2exbii 1730 |
. 2
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21 | 8, 11, 20 | 3bitri 279 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-fo 5607 df-fv 5609 df-1st 6820 df-2nd 6821 df-txp 30669 df-pprod 30670 |
This theorem is referenced by: brpprod3b 30703 brapply 30754 dfrdg4 30767 |
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