| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > 6p5lem | Structured version Visualization version GIF version | ||
| Description: Lemma for 6p5e11 12806 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 7442 | . 2 ⊢ (𝐴 + 𝐵) = (𝐴 + (𝐷 + 1)) |
| 3 | 6p5lem.1 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
| 4 | 3 | nn0cni 12538 | . . 3 ⊢ 𝐴 ∈ ℂ |
| 5 | 6p5lem.2 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
| 6 | 5 | nn0cni 12538 | . . 3 ⊢ 𝐷 ∈ ℂ |
| 7 | ax-1cn 11213 | . . 3 ⊢ 1 ∈ ℂ | |
| 8 | 4, 6, 7 | addassi 11271 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = (𝐴 + (𝐷 + 1)) |
| 9 | 1nn0 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 10 | 6p5lem.3 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
| 11 | 6p5lem.5 | . . . 4 ⊢ 𝐶 = (𝐸 + 1) | |
| 12 | 11 | eqcomi 2746 | . . 3 ⊢ (𝐸 + 1) = 𝐶 |
| 13 | 6p5lem.6 | . . 3 ⊢ (𝐴 + 𝐷) = ;1𝐸 | |
| 14 | 9, 10, 12, 13 | decsuc 12764 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = ;1𝐶 |
| 15 | 2, 8, 14 | 3eqtr2i 2771 | 1 ⊢ (𝐴 + 𝐵) = ;1𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 ℕ0cn0 12526 ;cdc 12733 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 |
| This theorem is referenced by: 6p5e11 12806 6p6e12 12807 7p4e11 12809 7p5e12 12810 7p6e13 12811 7p7e14 12812 8p3e11 12814 8p4e12 12815 8p5e13 12816 8p6e14 12817 8p7e15 12818 8p8e16 12819 9p2e11 12820 9p3e12 12821 9p4e13 12822 9p5e14 12823 9p6e15 12824 9p7e16 12825 9p8e17 12826 9p9e18 12827 |
| Copyright terms: Public domain | W3C validator |