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Mirrors > Home > MPE Home > Th. List > omsuc | Structured version Visualization version GIF version |
Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
omsuc | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgsuc 8054 | . . 3 ⊢ (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))) |
3 | suceloni 7522 | . . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On) | |
4 | omv 8131 | . . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵)) | |
5 | 3, 4 | sylan2 594 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵)) |
6 | ovex 7183 | . . . 4 ⊢ (𝐴 ·o 𝐵) ∈ V | |
7 | oveq1 7157 | . . . . 5 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴)) | |
8 | eqid 2821 | . . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) | |
9 | ovex 7183 | . . . . 5 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V | |
10 | 7, 8, 9 | fvmpt 6762 | . . . 4 ⊢ ((𝐴 ·o 𝐵) ∈ V → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝐴 ·o 𝐵) +o 𝐴)) |
11 | 6, 10 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝐴 ·o 𝐵) +o 𝐴) |
12 | omv 8131 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵)) | |
13 | 12 | fveq2d 6668 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))) |
14 | 11, 13 | syl5eqr 2870 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) +o 𝐴) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))) |
15 | 2, 5, 14 | 3eqtr4d 2866 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ↦ cmpt 5138 Oncon0 6185 suc csuc 6187 ‘cfv 6349 (class class class)co 7150 reccrdg 8039 +o coa 8093 ·o comu 8094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-omul 8101 |
This theorem is referenced by: omcl 8155 om0r 8158 om1r 8163 omordi 8186 omwordri 8192 omlimcl 8198 odi 8199 omass 8200 oneo 8201 omeulem1 8202 omeulem2 8203 oeoelem 8218 oaabs2 8266 omxpenlem 8612 cantnflt 9129 cantnflem1d 9145 infxpenc 9438 |
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