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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 15716 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | cchhllem.4 | . . . . 5 ⊢ (𝑁 < 5 ∨ 8 < 𝑁) | |
5 | 5lt8 11094 | . . . . . . . . 9 ⊢ 5 < 8 | |
6 | 2 | nnrei 10906 | . . . . . . . . . 10 ⊢ 𝑁 ∈ ℝ |
7 | 5re 10976 | . . . . . . . . . 10 ⊢ 5 ∈ ℝ | |
8 | 8re 10982 | . . . . . . . . . 10 ⊢ 8 ∈ ℝ | |
9 | 6, 7, 8 | lttri 10042 | . . . . . . . . 9 ⊢ ((𝑁 < 5 ∧ 5 < 8) → 𝑁 < 8) |
10 | 5, 9 | mpan2 703 | . . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 < 8) |
11 | 6, 8 | ltnei 10040 | . . . . . . . 8 ⊢ (𝑁 < 8 → 8 ≠ 𝑁) |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑁 < 5 → 8 ≠ 𝑁) |
13 | 12 | necomd 2837 | . . . . . 6 ⊢ (𝑁 < 5 → 𝑁 ≠ 8) |
14 | 8, 6 | ltnei 10040 | . . . . . 6 ⊢ (8 < 𝑁 → 𝑁 ≠ 8) |
15 | 13, 14 | jaoi 393 | . . . . 5 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 8) |
16 | 4, 15 | ax-mp 5 | . . . 4 ⊢ 𝑁 ≠ 8 |
17 | 1, 2 | ndxarg 15715 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
18 | ipndx 15845 | . . . . 5 ⊢ (·𝑖‘ndx) = 8 | |
19 | 17, 18 | neeq12i 2848 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (·𝑖‘ndx) ↔ 𝑁 ≠ 8) |
20 | 16, 19 | mpbir 220 | . . 3 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx) |
21 | 3, 20 | setsnid 15743 | . 2 ⊢ (𝐸‘((subringAlg ‘ℂfld)‘ℝ)) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
22 | eqidd 2611 | . . . 4 ⊢ (⊤ → ((subringAlg ‘ℂfld)‘ℝ) = ((subringAlg ‘ℂfld)‘ℝ)) | |
23 | ax-resscn 9872 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
24 | cnfldbas 19571 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
25 | 23, 24 | sseqtri 3600 | . . . . 5 ⊢ ℝ ⊆ (Base‘ℂfld) |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ (Base‘ℂfld)) |
27 | 22, 26, 1, 2, 4 | sralem 18998 | . . 3 ⊢ (⊤ → (𝐸‘ℂfld) = (𝐸‘((subringAlg ‘ℂfld)‘ℝ))) |
28 | 27 | trud 1484 | . 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 6106 | . 2 ⊢ (𝐸‘𝐶) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
31 | 21, 28, 30 | 3eqtr4i 2642 | 1 ⊢ (𝐸‘ℂfld) = (𝐸‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℂcc 9813 ℝcr 9814 · cmul 9820 < clt 9953 ℕcn 10897 5c5 10950 8c8 10953 ∗ccj 13684 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 Basecbs 15695 ·𝑖cip 15773 subringAlg csra 18989 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-sra 18993 df-cnfld 19568 |
This theorem is referenced by: (None) |
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