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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 16691 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | cchhllem.4 | . . . . 5 ⊢ (𝑁 < 5 ∨ 8 < 𝑁) | |
5 | 5lt8 11989 | . . . . . . . . 9 ⊢ 5 < 8 | |
6 | 2 | nnrei 11804 | . . . . . . . . . 10 ⊢ 𝑁 ∈ ℝ |
7 | 5re 11882 | . . . . . . . . . 10 ⊢ 5 ∈ ℝ | |
8 | 8re 11891 | . . . . . . . . . 10 ⊢ 8 ∈ ℝ | |
9 | 6, 7, 8 | lttri 10923 | . . . . . . . . 9 ⊢ ((𝑁 < 5 ∧ 5 < 8) → 𝑁 < 8) |
10 | 5, 9 | mpan2 691 | . . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 < 8) |
11 | 6, 8 | ltnei 10921 | . . . . . . . 8 ⊢ (𝑁 < 8 → 8 ≠ 𝑁) |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑁 < 5 → 8 ≠ 𝑁) |
13 | 12 | necomd 2987 | . . . . . 6 ⊢ (𝑁 < 5 → 𝑁 ≠ 8) |
14 | 8, 6 | ltnei 10921 | . . . . . 6 ⊢ (8 < 𝑁 → 𝑁 ≠ 8) |
15 | 13, 14 | jaoi 857 | . . . . 5 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 8) |
16 | 4, 15 | ax-mp 5 | . . . 4 ⊢ 𝑁 ≠ 8 |
17 | 1, 2 | ndxarg 16690 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
18 | ipndx 16825 | . . . . 5 ⊢ (·𝑖‘ndx) = 8 | |
19 | 17, 18 | neeq12i 2998 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (·𝑖‘ndx) ↔ 𝑁 ≠ 8) |
20 | 16, 19 | mpbir 234 | . . 3 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx) |
21 | 3, 20 | setsnid 16720 | . 2 ⊢ (𝐸‘((subringAlg ‘ℂfld)‘ℝ)) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
22 | eqidd 2737 | . . . 4 ⊢ (⊤ → ((subringAlg ‘ℂfld)‘ℝ) = ((subringAlg ‘ℂfld)‘ℝ)) | |
23 | ax-resscn 10751 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
24 | cnfldbas 20321 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
25 | 23, 24 | sseqtri 3923 | . . . . 5 ⊢ ℝ ⊆ (Base‘ℂfld) |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ (Base‘ℂfld)) |
27 | 22, 26, 1, 2, 4 | sralem 20168 | . . 3 ⊢ (⊤ → (𝐸‘ℂfld) = (𝐸‘((subringAlg ‘ℂfld)‘ℝ))) |
28 | 27 | mptru 1550 | . 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 6698 | . 2 ⊢ (𝐸‘𝐶) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
31 | 21, 28, 30 | 3eqtr4i 2769 | 1 ⊢ (𝐸‘ℂfld) = (𝐸‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 847 = wceq 1543 ⊤wtru 1544 ∈ wcel 2112 ≠ wne 2932 ⊆ wss 3853 〈cop 4533 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 ℂcc 10692 ℝcr 10693 · cmul 10699 < clt 10832 ℕcn 11795 5c5 11853 8c8 11856 ∗ccj 14624 ndxcnx 16663 sSet csts 16664 Slot cslot 16665 Basecbs 16666 ·𝑖cip 16754 subringAlg csra 20159 ℂfldccnfld 20317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-sra 20163 df-cnfld 20318 |
This theorem is referenced by: (None) |
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