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Mirrors > Home > ILE Home > Th. List > 6p5lem | GIF version |
Description: Lemma for 6p5e11 8417 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 5523 | . 2 ⊢ (𝐴 + 𝐵) = (𝐴 + (𝐷 + 1)) |
3 | 6p5lem.1 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 8193 | . . 3 ⊢ 𝐴 ∈ ℂ |
5 | 6p5lem.2 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
6 | 5 | nn0cni 8193 | . . 3 ⊢ 𝐷 ∈ ℂ |
7 | ax-1cn 6977 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | addassi 7035 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = (𝐴 + (𝐷 + 1)) |
9 | 1nn0 8197 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 6p5lem.3 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
11 | 6p5lem.5 | . . . 4 ⊢ 𝐶 = (𝐸 + 1) | |
12 | 11 | eqcomi 2044 | . . 3 ⊢ (𝐸 + 1) = 𝐶 |
13 | 6p5lem.6 | . . 3 ⊢ (𝐴 + 𝐷) = ;1𝐸 | |
14 | 9, 10, 12, 13 | decsuc 8390 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = ;1𝐶 |
15 | 2, 8, 14 | 3eqtr2i 2066 | 1 ⊢ (𝐴 + 𝐵) = ;1𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 (class class class)co 5512 1c1 6890 + caddc 6892 ℕ0cn0 8181 ;cdc 8368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-5 7976 df-6 7977 df-7 7978 df-8 7979 df-9 7980 df-10 7981 df-n0 8182 df-dec 8369 |
This theorem is referenced by: 6p5e11 8417 6p6e12 8418 7p4e11 8419 7p5e12 8420 7p6e13 8421 7p7e14 8422 8p3e11 8423 8p4e12 8424 8p5e13 8425 8p6e14 8426 8p7e15 8427 8p8e16 8428 9p2e11 8429 9p3e12 8430 9p4e13 8431 9p5e14 8432 9p6e15 8433 9p7e16 8434 9p8e17 8435 9p9e18 8436 |
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