Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > seqomeq12 | Structured version Visualization version GIF version |
Description: Equality theorem for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
Ref | Expression |
---|---|
seqomeq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7141 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏)) | |
2 | 1 | opeq2d 4772 | . . . . 5 ⊢ (𝐴 = 𝐵 → 〈suc 𝑎, (𝑎𝐴𝑏)〉 = 〈suc 𝑎, (𝑎𝐵𝑏)〉) |
3 | 2 | mpoeq3dv 7212 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉)) |
4 | fveq2 6645 | . . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷)) | |
5 | 4 | opeq2d 4772 | . . . 4 ⊢ (𝐶 = 𝐷 → 〈∅, ( I ‘𝐶)〉 = 〈∅, ( I ‘𝐷)〉) |
6 | rdgeq12 8032 | . . . 4 ⊢ (((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉) ∧ 〈∅, ( I ‘𝐶)〉 = 〈∅, ( I ‘𝐷)〉) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) = rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉)) | |
7 | 3, 5, 6 | syl2an 598 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) = rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉)) |
8 | 7 | imaeq1d 5895 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) “ ω) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉) “ ω)) |
9 | df-seqom 8067 | . 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) “ ω) | |
10 | df-seqom 8067 | . 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉) “ ω) | |
11 | 8, 9, 10 | 3eqtr4g 2858 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Vcvv 3441 ∅c0 4243 〈cop 4531 I cid 5424 “ cima 5522 suc csuc 6161 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ωcom 7560 reccrdg 8028 seqωcseqom 8066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-iota 6283 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seqom 8067 |
This theorem is referenced by: cantnffval 9110 cantnfval 9115 cantnfres 9124 cnfcomlem 9146 cnfcom2 9149 fin23lem33 9756 |
Copyright terms: Public domain | W3C validator |