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Mirrors > Home > MPE Home > Th. List > cchhllem | Structured version Visualization version GIF version |
Description: Lemma for chlbas and chlvsca . (Contributed by Thierry Arnoux, 15-Apr-2019.) |
Ref | Expression |
---|---|
cchhl.c | ⊢ 𝐶 = (((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉) |
cchhllem.2 | ⊢ 𝐸 = Slot 𝑁 |
cchhllem.3 | ⊢ 𝑁 ∈ ℕ |
cchhllem.4 | ⊢ (𝑁 < 5 ∨ 8 < 𝑁) |
Ref | Expression |
---|---|
cchhllem | ⊢ (𝐸‘ℂfld) = (𝐸‘𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cchhllem.2 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | cchhllem.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 16365 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | cchhllem.4 | . . . . 5 ⊢ (𝑁 < 5 ∨ 8 < 𝑁) | |
5 | 5lt8 11641 | . . . . . . . . 9 ⊢ 5 < 8 | |
6 | 2 | nnrei 11449 | . . . . . . . . . 10 ⊢ 𝑁 ∈ ℝ |
7 | 5re 11529 | . . . . . . . . . 10 ⊢ 5 ∈ ℝ | |
8 | 8re 11541 | . . . . . . . . . 10 ⊢ 8 ∈ ℝ | |
9 | 6, 7, 8 | lttri 10566 | . . . . . . . . 9 ⊢ ((𝑁 < 5 ∧ 5 < 8) → 𝑁 < 8) |
10 | 5, 9 | mpan2 678 | . . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 < 8) |
11 | 6, 8 | ltnei 10564 | . . . . . . . 8 ⊢ (𝑁 < 8 → 8 ≠ 𝑁) |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑁 < 5 → 8 ≠ 𝑁) |
13 | 12 | necomd 3023 | . . . . . 6 ⊢ (𝑁 < 5 → 𝑁 ≠ 8) |
14 | 8, 6 | ltnei 10564 | . . . . . 6 ⊢ (8 < 𝑁 → 𝑁 ≠ 8) |
15 | 13, 14 | jaoi 843 | . . . . 5 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 8) |
16 | 4, 15 | ax-mp 5 | . . . 4 ⊢ 𝑁 ≠ 8 |
17 | 1, 2 | ndxarg 16364 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
18 | ipndx 16497 | . . . . 5 ⊢ (·𝑖‘ndx) = 8 | |
19 | 17, 18 | neeq12i 3034 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (·𝑖‘ndx) ↔ 𝑁 ≠ 8) |
20 | 16, 19 | mpbir 223 | . . 3 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx) |
21 | 3, 20 | setsnid 16395 | . 2 ⊢ (𝐸‘((subringAlg ‘ℂfld)‘ℝ)) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
22 | eqidd 2780 | . . . 4 ⊢ (⊤ → ((subringAlg ‘ℂfld)‘ℝ) = ((subringAlg ‘ℂfld)‘ℝ)) | |
23 | ax-resscn 10392 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
24 | cnfldbas 20251 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
25 | 23, 24 | sseqtri 3894 | . . . . 5 ⊢ ℝ ⊆ (Base‘ℂfld) |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ (Base‘ℂfld)) |
27 | 22, 26, 1, 2, 4 | sralem 19671 | . . 3 ⊢ (⊤ → (𝐸‘ℂfld) = (𝐸‘((subringAlg ‘ℂfld)‘ℝ))) |
28 | 27 | mptru 1514 | . 2 ⊢ (𝐸‘ℂfld) = (𝐸‘((subringAlg ‘ℂfld)‘ℝ)) |
29 | cchhl.c | . . 3 ⊢ 𝐶 = (((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉) | |
30 | 29 | fveq2i 6502 | . 2 ⊢ (𝐸‘𝐶) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
31 | 21, 28, 30 | 3eqtr4i 2813 | 1 ⊢ (𝐸‘ℂfld) = (𝐸‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 833 = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 ≠ wne 2968 ⊆ wss 3830 〈cop 4447 class class class wbr 4929 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 ℂcc 10333 ℝcr 10334 · cmul 10340 < clt 10474 ℕcn 11439 5c5 11498 8c8 11501 ∗ccj 14316 ndxcnx 16336 sSet csts 16337 Slot cslot 16338 Basecbs 16339 ·𝑖cip 16426 subringAlg csra 19662 ℂfldccnfld 20247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-sra 19666 df-cnfld 20248 |
This theorem is referenced by: (None) |
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