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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 16511 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | cchhllem.4 | . . . . 5 ⊢ (𝑁 < 5 ∨ 8 < 𝑁) | |
5 | 5lt8 11834 | . . . . . . . . 9 ⊢ 5 < 8 | |
6 | 2 | nnrei 11649 | . . . . . . . . . 10 ⊢ 𝑁 ∈ ℝ |
7 | 5re 11727 | . . . . . . . . . 10 ⊢ 5 ∈ ℝ | |
8 | 8re 11736 | . . . . . . . . . 10 ⊢ 8 ∈ ℝ | |
9 | 6, 7, 8 | lttri 10768 | . . . . . . . . 9 ⊢ ((𝑁 < 5 ∧ 5 < 8) → 𝑁 < 8) |
10 | 5, 9 | mpan2 689 | . . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 < 8) |
11 | 6, 8 | ltnei 10766 | . . . . . . . 8 ⊢ (𝑁 < 8 → 8 ≠ 𝑁) |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑁 < 5 → 8 ≠ 𝑁) |
13 | 12 | necomd 3073 | . . . . . 6 ⊢ (𝑁 < 5 → 𝑁 ≠ 8) |
14 | 8, 6 | ltnei 10766 | . . . . . 6 ⊢ (8 < 𝑁 → 𝑁 ≠ 8) |
15 | 13, 14 | jaoi 853 | . . . . 5 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 8) |
16 | 4, 15 | ax-mp 5 | . . . 4 ⊢ 𝑁 ≠ 8 |
17 | 1, 2 | ndxarg 16510 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
18 | ipndx 16643 | . . . . 5 ⊢ (·𝑖‘ndx) = 8 | |
19 | 17, 18 | neeq12i 3084 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (·𝑖‘ndx) ↔ 𝑁 ≠ 8) |
20 | 16, 19 | mpbir 233 | . . 3 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx) |
21 | 3, 20 | setsnid 16541 | . 2 ⊢ (𝐸‘((subringAlg ‘ℂfld)‘ℝ)) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
22 | eqidd 2824 | . . . 4 ⊢ (⊤ → ((subringAlg ‘ℂfld)‘ℝ) = ((subringAlg ‘ℂfld)‘ℝ)) | |
23 | ax-resscn 10596 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
24 | cnfldbas 20551 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
25 | 23, 24 | sseqtri 4005 | . . . . 5 ⊢ ℝ ⊆ (Base‘ℂfld) |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ (Base‘ℂfld)) |
27 | 22, 26, 1, 2, 4 | sralem 19951 | . . 3 ⊢ (⊤ → (𝐸‘ℂfld) = (𝐸‘((subringAlg ‘ℂfld)‘ℝ))) |
28 | 27 | mptru 1544 | . 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 6675 | . 2 ⊢ (𝐸‘𝐶) = (𝐸‘(((subringAlg ‘ℂfld)‘ℝ) sSet 〈(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))〉)) |
31 | 21, 28, 30 | 3eqtr4i 2856 | 1 ⊢ (𝐸‘ℂfld) = (𝐸‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ℂcc 10537 ℝcr 10538 · cmul 10544 < clt 10677 ℕcn 11640 5c5 11698 8c8 11701 ∗ccj 14457 ndxcnx 16482 sSet csts 16483 Slot cslot 16484 Basecbs 16485 ·𝑖cip 16572 subringAlg csra 19942 ℂfldccnfld 20547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-sra 19946 df-cnfld 20548 |
This theorem is referenced by: (None) |
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