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Mirrors > Home > ILE Home > Th. List > 6p5lem | GIF version |
Description: Lemma for 6p5e11 9415 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
6p5lem.1 | ⊢ 𝐴 ∈ ℕ0 |
6p5lem.2 | ⊢ 𝐷 ∈ ℕ0 |
6p5lem.3 | ⊢ 𝐸 ∈ ℕ0 |
6p5lem.4 | ⊢ 𝐵 = (𝐷 + 1) |
6p5lem.5 | ⊢ 𝐶 = (𝐸 + 1) |
6p5lem.6 | ⊢ (𝐴 + 𝐷) = ;1𝐸 |
Ref | Expression |
---|---|
6p5lem | ⊢ (𝐴 + 𝐵) = ;1𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6p5lem.4 | . . 3 ⊢ 𝐵 = (𝐷 + 1) | |
2 | 1 | oveq2i 5864 | . 2 ⊢ (𝐴 + 𝐵) = (𝐴 + (𝐷 + 1)) |
3 | 6p5lem.1 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 9147 | . . 3 ⊢ 𝐴 ∈ ℂ |
5 | 6p5lem.2 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
6 | 5 | nn0cni 9147 | . . 3 ⊢ 𝐷 ∈ ℂ |
7 | ax-1cn 7867 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | addassi 7928 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = (𝐴 + (𝐷 + 1)) |
9 | 1nn0 9151 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 6p5lem.3 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
11 | 6p5lem.5 | . . . 4 ⊢ 𝐶 = (𝐸 + 1) | |
12 | 11 | eqcomi 2174 | . . 3 ⊢ (𝐸 + 1) = 𝐶 |
13 | 6p5lem.6 | . . 3 ⊢ (𝐴 + 𝐷) = ;1𝐸 | |
14 | 9, 10, 12, 13 | decsuc 9373 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = ;1𝐶 |
15 | 2, 8, 14 | 3eqtr2i 2197 | 1 ⊢ (𝐴 + 𝐵) = ;1𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 (class class class)co 5853 1c1 7775 + caddc 7777 ℕ0cn0 9135 ;cdc 9343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-dec 9344 |
This theorem is referenced by: 6p5e11 9415 6p6e12 9416 7p4e11 9418 7p5e12 9419 7p6e13 9420 7p7e14 9421 8p3e11 9423 8p4e12 9424 8p5e13 9425 8p6e14 9426 8p7e15 9427 8p8e16 9428 9p2e11 9429 9p3e12 9430 9p4e13 9431 9p5e14 9432 9p6e15 9433 9p7e16 9434 9p8e17 9435 9p9e18 9436 |
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