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Mirrors > Home > NFE Home > Th. List > tfin1c | Unicode version |
Description: The finite T operator is idempotent over 1c. Theorem X.1.34(a) of [Rosser] p. 529. (Contributed by SF, 30-Jan-2015.) |
Ref | Expression |
---|---|
tfin1c | Tfin 1c 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 4402 | . . 3 0c Nn | |
2 | addcid2 4407 | . . . 4 0c 1c 1c | |
3 | 1cex 4142 | . . . . . 6 1c | |
4 | 3 | snel1c 4140 | . . . . 5 1c 1c |
5 | ne0i 3556 | . . . . 5 1c 1c 1c | |
6 | 4, 5 | ax-mp 5 | . . . 4 1c |
7 | 2, 6 | eqnetri 2533 | . . 3 0c 1c |
8 | tfinsuc 4498 | . . 3 0c Nn 0c 1c Tfin 0c 1c Tfin 0c 1c | |
9 | 1, 7, 8 | mp2an 653 | . 2 Tfin 0c 1c Tfin 0c 1c |
10 | tfineq 4488 | . . 3 0c 1c 1c Tfin 0c 1c Tfin 1c | |
11 | 2, 10 | ax-mp 5 | . 2 Tfin 0c 1c Tfin 1c |
12 | tfin0c 4497 | . . . 4 Tfin 0c 0c | |
13 | 12 | addceq1i 4386 | . . 3 Tfin 0c 1c 0c 1c |
14 | 13, 2 | eqtri 2373 | . 2 Tfin 0c 1c 1c |
15 | 9, 11, 14 | 3eqtr3i 2381 | 1 Tfin 1c 1c |
Colors of variables: wff setvar class |
Syntax hints: wceq 1642 wcel 1710 wne 2516 c0 3550 csn 3737 1cc1c 4134 Nn cnnc 4373 0cc0c 4374 cplc 4375 Tfin ctfin 4435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-tfin 4443 |
This theorem is referenced by: oddtfin 4518 sfintfin 4532 |
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