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Mirrors > Home > MPE Home > Th. List > 6p5lem | Structured version Visualization version GIF version |
Description: Lemma for 6p5e11 12804 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 12536 | . . 3 ⊢ 𝐴 ∈ ℂ |
5 | 6p5lem.2 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
6 | 5 | nn0cni 12536 | . . 3 ⊢ 𝐷 ∈ ℂ |
7 | ax-1cn 11211 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | addassi 11269 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = (𝐴 + (𝐷 + 1)) |
9 | 1nn0 12540 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 6p5lem.3 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
11 | 6p5lem.5 | . . . 4 ⊢ 𝐶 = (𝐸 + 1) | |
12 | 11 | eqcomi 2744 | . . 3 ⊢ (𝐸 + 1) = 𝐶 |
13 | 6p5lem.6 | . . 3 ⊢ (𝐴 + 𝐷) = ;1𝐸 | |
14 | 9, 10, 12, 13 | decsuc 12762 | . 2 ⊢ ((𝐴 + 𝐷) + 1) = ;1𝐶 |
15 | 2, 8, 14 | 3eqtr2i 2769 | 1 ⊢ (𝐴 + 𝐵) = ;1𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 1c1 11154 + caddc 11156 ℕ0cn0 12524 ;cdc 12731 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12732 |
This theorem is referenced by: 6p5e11 12804 6p6e12 12805 7p4e11 12807 7p5e12 12808 7p6e13 12809 7p7e14 12810 8p3e11 12812 8p4e12 12813 8p5e13 12814 8p6e14 12815 8p7e15 12816 8p8e16 12817 9p2e11 12818 9p3e12 12819 9p4e13 12820 9p5e14 12821 9p6e15 12822 9p7e16 12823 9p8e17 12824 9p9e18 12825 |
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