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Mirrors > Home > ILE Home > Th. List > fodjumkvlemres | Unicode version |
Description: Lemma for fodjumkv 7115. The final result with expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.) |
Ref | Expression |
---|---|
fodjumkv.o | Markov |
fodjumkv.fo | ⊔ |
fodjumkv.p | inl |
Ref | Expression |
---|---|
fodjumkvlemres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodjumkv.fo | . . . . . 6 ⊔ | |
2 | 1 | adantr 274 | . . . . 5 ⊔ |
3 | fodjumkv.p | . . . . 5 inl | |
4 | simpr 109 | . . . . 5 | |
5 | 2, 3, 4 | fodju0 7102 | . . . 4 |
6 | 5 | ex 114 | . . 3 |
7 | 6 | necon3ad 2376 | . 2 |
8 | fveq1 5479 | . . . . . . 7 | |
9 | 8 | eqeq1d 2173 | . . . . . 6 |
10 | 9 | ralbidv 2464 | . . . . 5 |
11 | 10 | notbid 657 | . . . 4 |
12 | 8 | eqeq1d 2173 | . . . . 5 |
13 | 12 | rexbidv 2465 | . . . 4 |
14 | 11, 13 | imbi12d 233 | . . 3 |
15 | fodjumkv.o | . . . 4 Markov | |
16 | ismkvmap 7109 | . . . . 5 Markov Markov | |
17 | 16 | ibi 175 | . . . 4 Markov |
18 | 15, 17 | syl 14 | . . 3 |
19 | 1, 3, 15 | fodjuf 7100 | . . 3 |
20 | 14, 18, 19 | rspcdva 2830 | . 2 |
21 | 1 | adantr 274 | . . . 4 ⊔ |
22 | simpr 109 | . . . . 5 | |
23 | fveqeq2 5489 | . . . . . 6 | |
24 | 23 | cbvrexv 2690 | . . . . 5 |
25 | 22, 24 | sylib 121 | . . . 4 |
26 | 21, 3, 25 | fodjum 7101 | . . 3 |
27 | 26 | ex 114 | . 2 |
28 | 7, 20, 27 | 3syld 57 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1342 wex 1479 wcel 2135 wne 2334 wral 2442 wrex 2443 c0 3404 cif 3515 cmpt 4037 wfo 5180 cfv 5182 (class class class)co 5836 c1o 6368 c2o 6369 cmap 6605 ⊔ cdju 6993 inlcinl 7001 Markovcmarkov 7106 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-1o 6375 df-2o 6376 df-map 6607 df-dju 6994 df-inl 7003 df-inr 7004 df-markov 7107 |
This theorem is referenced by: fodjumkv 7115 |
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