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Mirrors > Home > MPE Home > Th. List > 6p5lem | Structured version Visualization version GIF version |
Description: Lemma for 6p5e11 12172 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 7167 | . 2 ⊢ (𝐴 + 𝐵) = (𝐴 + (𝐷 + 1)) |
3 | 6p5lem.1 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 11910 | . . 3 ⊢ 𝐴 ∈ ℂ |
5 | 6p5lem.2 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
6 | 5 | nn0cni 11910 | . . 3 ⊢ 𝐷 ∈ ℂ |
7 | ax-1cn 10595 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | addassi 10651 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = (𝐴 + (𝐷 + 1)) |
9 | 1nn0 11914 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 6p5lem.3 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
11 | 6p5lem.5 | . . . 4 ⊢ 𝐶 = (𝐸 + 1) | |
12 | 11 | eqcomi 2830 | . . 3 ⊢ (𝐸 + 1) = 𝐶 |
13 | 6p5lem.6 | . . 3 ⊢ (𝐴 + 𝐷) = ;1𝐸 | |
14 | 9, 10, 12, 13 | decsuc 12130 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = ;1𝐶 |
15 | 2, 8, 14 | 3eqtr2i 2850 | 1 ⊢ (𝐴 + 𝐵) = ;1𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 1c1 10538 + caddc 10540 ℕ0cn0 11898 ;cdc 12099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 |
This theorem is referenced by: 6p5e11 12172 6p6e12 12173 7p4e11 12175 7p5e12 12176 7p6e13 12177 7p7e14 12178 8p3e11 12180 8p4e12 12181 8p5e13 12182 8p6e14 12183 8p7e15 12184 8p8e16 12185 9p2e11 12186 9p3e12 12187 9p4e13 12188 9p5e14 12189 9p6e15 12190 9p7e16 12191 9p8e17 12192 9p9e18 12193 |
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